I can't say whether any of the votes for Michael Romanov had anything to do with his "Roman" surname.
I do have some comments on the Third Rome ideologies of Russia and other states. And I have a suggestion for how the Romanov family could have claimed to be descended from Roman Emperors, though as far as I know the Romanovs never did.
Actually the idea of a Third Rome originated in Bulgaria in the 14th Century (1301-1400) as far as I know.
The Bulgarian "third Rome" was naturally the capital of the second Bulgarian Empire, Veliko Tarnovo, which unfortunately for the "third Rome" theory, fell to the Ottoman Turks on 17 July 1393, 60 years before Constantinople, "the second Rome" fell to the Turks on 29 May 1453.
And I have to say that the weak and feeble claim of the Second Bulgarian Empire to be one Roman Empire out of several, was far stronger than that of the Russian Tsars. At least the Bulgarian Tsars used the title of "Emperor and autocrat of the Bulgarians and the Romans". I don't find that title very impressive.
Masties, ruler of the Kingdom of the Aures from c.426-494 or from 449-516, is said to have used the title of "Emperor of the Romans and the Berbers" for the last 40 or 10 years of his reign.
And at least Masties put Romans first before Berbers. That is more respectful of the Roman Empire than putting Berbers first would be, and and so the title of Masties indicates that Masties was closer to being a genuine Roman Emperor than a Bulgarian "Emperor of the bulgarians and the Romans" was, or than Stefan Dusan, "Emperor of the Serbs and the Romans, the Bulgarians and the Albanians" was.
Going by title alone, one might suppose that Ottoman sultans and padishahs who used the title of Kayser-i-Rum, "Caesar of Rome", were more Roman than Masties, "Emperor of the Romaans and the Berbers" who in turn was more Roman than any "Emperor of the Bulgarians and the Romans".
But from the very beginning of the Ottoman state the Ottomans were enemies of the Roman Empire, raiding Roman lands and killing Romans, and stealing property from Romans, and stealing Romans themselves as slaves, and sometimes launching large invasions to conquer lands from the Roman Empire.
The Ottomans were always foreign and external enemies of the Roman Empire, and no Ottoman could ever become a rightful Roman Emperor.
And the Ottoman rulers used many other titles much more prominently than Kaysar-i-Rum, such as Sultan, Khan, and Padishah. If they really considered themselves Roman Emperors and thus the rightful rulers of all the world, why would they use titles from other and rival political traditions?
And how does that relate to the Russian Tsars? How did the title of the Russian Tsars compare to that of Masties, Bulgarian and Serbian rulers, and Ottoman Sultans?
Very poorly. As far as I can tell, no Russian ruler ever claimed to be a Roman Emperor in their title. As far as I know, no Russian ruler ever officially renamed their realm the Roman Empire.
If the Russian rulers never actually said that their realm was the Roman Empire, and if they never actually said they were Roman Emperors, where is the claim that Russia was the heir of the eastern Roman or "Byzantine" Empire? Some people may have claimed that Moscow should be "the Third Rome" and that Russia was the rightful heir of the Roman Empire, but no Russian ruler considered such claims important and desirable enough to make them official claims of his government.
So I don't consider the claims of Moscow as the third Rome to be very significant, certainly not significant enough to bother examining for validity.
Here is a link to a list of claims of Roman Heritage and to be the Third Rome:
As far as I am concerned no mere nation can ever have a rightful claim to be the rightful heir of any empire. If a state renames itself from a nation to an empire, adopts an imperial ideology instead of a nationalistic ideology, and teaches its citizens to stop being patriotic for a nation and to start being imperialistic for the empire, then it may become an empire and may thus be eligible to (possibly) be the heir or continuation of some specific previous empire.
And as far as I am concerned, no state can qualify to be considered to be the heir, or even one of several heirs, of the Roman empire until it renames itself the Roman Empire and until its head of state takes the title of Imperator Caesar Augustus, or Imperator Romanorum et semper Augustus, or Baselius kai Autokrator ton Rhomaion. And then, it will merely have qualified to have its claim to be the Roman Empire considered and judged, not to be automatically accepted as the Roman Empire.
To answer the original question, it seems clear to me that the Russian political elite didn't take the Third Rome claim seriously enough to change the name of the state or change the title of the ruler. And certainly they knew that Michael's surname "Romanov" didn't mean "descendant of a Roman emigrant to Russia" but "descendant of a man with the personal name of Roman", who would have been named Roman because his father was a fan of things Roman, maybe, or thought that Roman sounded grand and high.
But I have never read any hypothetical minutes of the deliberations and election of Michael Romanov that could have been written. So I certainly can't rule out the possibility that some votes may have been cast for Michael Romanov because of his Roman surname.
A possible Romanov claim to have Roman ancestry, although I don't know if the Romanovs ever made such a claim.
The Rurik dynasty that ruled part or all of Russia from about 862 to 1610 was possibly of Scandinavian ancestry. It has been suggested that Rurik was probably Rorik of Dorestad who flourished in western Europe from about 840 to about 860, and 870 to 882 or earlier. Rurik was supposed to have come to Russia in 862 and died in 1879. Rorik of Dorestad and was probably the nephew of Harald Klak, king of part or all of Denmark from 812 to 813 and from 819 to 827, who in turn might have been the son of a Halfdan. And it is possible that there were legendary family trees tracing the ancestry of Rorik of Dorestad for many generations and centuries back.
However, it is certainly possible and maybe probable that Rurik was not Rorik of Dorestad and his ancestry may be totally unknown.
I believe that Ivan IV the Terrible (1530-1584) claimed that the Rurik Dynasty was descended from the Julio-Claudian Emperors. As I recall the claim, the first Emperor, Augustus, supposedly had a brother or nephew named Prusus (Augustus did have relatives named Drusus), and Augustus appointed Prusus the ruler of a region on the Baltic Sea. The people of that region were called Prussians (Old Prussians, not German Prussans) after Prusas. And about 800 years later an (Old) Prussian prince named Rurik, descended from Prusas, moved to Russia and founded the Rurik dynasty.
Many branches of the Rurik dynasty exist today, but the Romanovs are not a branch of the Rurik dynasty, so it has always been a bit of puzzle why Michael Romanov was elected Tsar instead of one of many Rurikid nobles and princes.
However, the Romanovs did claim to be descended though a female ancestor from a branch of the Ruik dynasty, which is not very improbable after all. You can find it here:
I think, but unfortunately I forget which of the many, many, many Rurik branches that descent is from.
[Added 03-14-2020. That line of descent from the House of Rukik is in post # 28 of the thread: https://historum.com/threads/heirs-of-the-kingdom-of-sweden.171986/page-34]
I did find one line of Rurikid descent:
Feodor Nikitich Romanov was descended from the Rurik dynasty through the female line. His mother, Evdokiya Gorbataya-Shuyskaya, was a Rurikid princess from the Shuysky branch, daughter of Alexander Gorbatyi-Shuisky.
So apparently the mother and the paternal grandmother of Tsar Michael I Romanov were both Rurikid princesses.
So now I turn to what is known about the ancestry of the Romanov Dynasty.
The Romanovs share their origin with two dozen other Russian noble families. Their earliest common ancestor is one Andrei Kobyla, attested around 1347 as a boyar in the service of Semyon I of Moscow.5 Later generations assigned to Kobyla an illustrious pedigree. An 18th-century genealogy claimed that he was the son of the Old Prussians prince Glanda Kambila, who came to Russia in the second half of the 13th century, fleeing the invading Germans. Indeed, one of the leaders of the Old Prussians rebellion of 1260–1274 against the Teutonic order was named Glande. This legendary version of the Romanov's origin is contested by another version of their descent from a boyar family from Novgorod.
His actual origin may have been less spectacular. Not only is Kobyla Russian for "mare", some of his relatives also had as nicknames the terms for horses and other domestic animals, thus suggesting descent from one of the royal equerries. One of Kobyla's sons, Feodor, a member of the boyar Duma of Dmitri Donskoi, was nicknamed Koshka ("cat"). His descendants took the surname Koshkin, then changed it to Zakharin, which family later split into two branches: Zakharin-Yakovlev and Zakharin-Yuriev.5 During the reign of Ivan the Terrible, the former family became known as Yakovlev (Alexander Herzen among them), whereas grandchildren of Roman Yurievich Zakharyin-Yuriev [ru] changed their name to "Romanov".5
So if the Romanovs claimed descent from an Old Prussian prince, they might have claimed to be distant cousins of the Rurikids who claimed that Rurik was an Old Prussian prince, descended from Prusas the relative of Emperor Augustus. Thus the Romanovs might have just as much (or as little) reason to claim to Roman ancestry as the Rurikids did.
But I don't know if any Romanovs ever claimed to be descended from Julio-Claudian
Roman Emperors or if that was a factor in electing Michael Romanov Tsar.
In his answer Pieter Geerkens says that;
The Russian orthodox Church became autocephalous in 1448, half a decade before Constantinople fell (though such was clearly imminent). Throughout the Middle Ages the true heir of the Roman Empire was seen to be the various Christian Catholic Patriarchies, of Rome, Constantinople, Alexandria, Antioch, and Jerusalem. It was as natural for the Russian Patriarch to claim right of successorship upon the fall of Constantinople as it was for the Ottoman Caliph to do so. (This was why for centuries Holy Roman Emperors were crowned by the Pope, and why Napoleon arranged likewise in 1804.) It was this claim, predating the ascension of Michael Romanov (in 1613) by 160 years, that underlay the concept of a Russian Empire at Moscow being the successors of the Roman Empire and Rome...
So Pieter Geerkens said:"Throughout the Middle Ages the true heir of the Roman Empire was seen to be the various Christian Catholic Patriarchies, of Rome, Constantinople, Alexandria, Antioch, and Jerusalem."
And I say that he is wrong, that it is logically impossible for a religious institution to become the heir and successor to the rights of a political institution.
Remember what Jesus said when asked whether it was right to pay taxes to the Roman Empire. "Render unto Caesar that which is Caesar's, and unto God that which is God's". That clearly said that God Almighty did not want or need the secular obedience that men paid to secular rulers, and thus that men who are, or who claim to be, representatives of God have no power or right to demand the secular obedience and taxes than men give to secular rulers.
It also said the Christians, and Christian clergy in particular, should not seek political power, because exerting secular political power, and even more so seeking to gain it, often involves behaving in a non Christian way that no good Christian should behave.
It also means that Christians should pay taxes to whoever has political power, right? Well, Christians should pay taxes to whoever has political power just as much as they should worship any god that happens to be worshiped where they live.
"Render unto God that which is God's" means render unto the Christian God, Yahweh/Jehovah/Allah the worship that which belongs to the Christian God, but don't render unto other gods, false gods, that which is God's. It means that it would be wrong for Christians to give any other alleged god that which is God's, belief and worship.
And it logically follows that the other part of the answer must be the same type of answer, it must mean that there is only one person that Christians must pay taxes to, the Emperor of the Roman Empire, and that all Christians must do so.
So even when the Roman empire was ruled by pagans, Jesus Christ said that all Christians must pay taxes to the Emperor. Even when the pagan local governor of the Roman Empire was pressured into sentencing Jesus to death, Jesus did not call on his divine side and use his infinite divine powers to stop his execution, setting an example of obedience to the Roman Empire even unto death for all his followers.
For centuries Christians who were executed for following an strange and suspicious religion calmly accepted their fates without attempting to fight or revolt against the Empire. Even the highest Christians, bishops and archbishops and the popes of Rome, accepted being deposed and imprisoned or executed by the imperial government, setting an example of loyalty to the emperor.
So when emperors converted to Christianity and a larger and larger proportion of the imperial population converted to Christianity, there was even less reason (if it is possible for reasons to be less than zero), for Christians in general, and for Christian clergy in particular, to revolt against the Empire or seek political power for themselves.
I note that when the western section of the Roman Empire crumbled in about AD 476 or 480, the eastern section of the Roman Empire continued as before. The eastern Roman or "Byzantine" emperors were the rightful successors and heirs to the political power of the western Roman Emperors.
And the Patriarchs of Constantinople, Antioch, Jerusalem, and Alexandria agreed with that view and were still loyal subjects of the Roman Empire. Only the Patriarchs of Rome were outside of the firm political rule of the eastern Roman Empire, and that fact did nothing to diminish their duty to be obedient subjects of the Emperor.
Pieter Geerkens wrote: "It was as natural for the Russian Patriarch to claim right of successorship upon the fall of Constantinople as it was for the Ottoman Caliph to do so."
Of course it is very natural for people to claim things that they don't have any right to.
The Ottoman emir, sultan, padishah, caliph, and khagan had absolutely no right to claim the imperial succession. For over 150 years from about 1300, when it is first recorded, to 1453, the Ottoman state was an enemy of the eastern Roman or "Byzantine" empire, constantly raiding it and often invading it and conquering lands from it. The Ottomans were enemies of Rome, and could never become Romans or Roman rulers.
When Constantinople fell in 1453, and Emperor Constantine XI was killed, the remaining region of the Empire, the Morea, had the right and duty to select a new emperor. But the two Despots distrusted each other too much to make one of them emperor, and the Ottomans conquered the Morea in 1460/61.
In 1460/61 the heirship and succession to the eastern Roman or "Byzantine" empire then may have passed to the Roman state with the closest lineage to it, the "Empire of Trebizond" or "Trapezuntine Empire". In 1282 the "Byzantine" emperor had persuaded the Trapezuntine ruler to abandon the title of "Emperor of the Romans" and to adopt the more modest title of "Emperor of All the East, of the Iberians, and of the Trans Marine Provinces".
So Emperor David Megas Komnenos had reason to call himself "Emperor of the Romans" starting in 1453 or 1460. But Trebizond was conquered by the Ottomans in 1461.
In 1461 the heirship and succession to the eastern Roman or "Byzantine" state may have then passed to the remaining roman state with the closest link, the Principality of Gothia or of Theodoro in the Crimea. But the Ottomans conquered that principality in 1475.
An alternate theory would be that the closest relatives of Emperor Constantine XI Palaiolologos were his rightful heirs. They were his two brothers, the Despots of the Morea, who were deposed in 1460.
Thomas Palaiologos moved to Italy, and was recognized in western Europe as the titular emperor until he died in 1465, and his son Andreas Palaiologos was recognized as emperor until he died with no known children in 1502.
So Thomas and Andreas may have become the heirs to the eastern Roman Empire in 1453, 1460, 1461, or 1475.
Andreas Palaiologos had a sister, Zoe/Sophia Palaiologina (c. 1440/49-1503), who married Ivan III (1440-1505), Great Prince of Russia. But their last known descendant was Maria Vladimirovna of Staritsa (c.1650-1610), who outlived her children.
Anyway, Zoe/Sophia's descendants never became the rightful Palaiologos heirs, since she had an older sister Helena Palaiologina (1431-1473), who married Lazar Brankovic (c. 1421-14558), Despot of Serbia. they had three daughters who married, and the heirs of two of those daughters are Luigi Serra, 11th Duke of Cassano (b. 1939) and Beneditto Francesco Barberini, Prince of Palestrina (b. 1961).
See post # 98 here: https://historum.com/threads/heirs-of-the-byzantine-empire.121359/page-106
So that accounts for the second alternative, genealogical succession.
There is a third alternative, Frederick III (1415-1493) became King of the Romans and Always Emperor in 1440 and was crowned Emperor of the Romans and Always Emperor in 1553, As a ruler of another Roman Empire Frederick III may have become the heir and successor of Constantine XI in 1453, 1460, 1461, or 1475, or the next emperor, Maximilian I, may have become the heir and successor of Constantine XI in 1502 when Andreas Palaiologos died with no known children.
Thus the succeeding rulers of the Holy Roman Emperors may have been the rightful heirs or successors of the eastern Roman or "Byzantine" emperors.
in any case, the idea that the rightful succession went to either the Palaiologos heirs or to the Holy Roman Emperors is far more logical and reasonable than the idea it went to either the Bishops of Moscow or to the Ottoman enemies of the Roman Empire.
I may also note that that there was no patriarch of Moscow or anywhere in Russia in 1453. The first Patriarch of Moscow and All Russia was appointed in 1589; the position was abolished in 1721 and recreated in 1917.