A claim to the imperial throne, however valid it may have been, was sold by a nephew of Emperor Constantine XI to King Charles VIII of France, and later willed to King & Queen Ferdinand and Isabella of spain.
According to Wikipedia, Thomas Palaiologos, co Despot of the Morea until the Turkish conquest in 1460, claimed the title of Emperor from 1460 until his death in 1465, and his son Andreas Palaiologos claimed the imperial title until his death in 1502 without any known children.
Desperate for money, Andreas sold his rights to the Byzantine crown in 1494 to Charles VIII of France, who attempted to organize a crusade against the Ottomans. The sale was conditional on Charles, who Andreas hoped to use as a champion against the Ottomans, conquering the Morea and granting it to Andreas. When Charles died in 1498, Andreas once again claimed the imperial titles, using them until his death. He died in poverty in Rome in 1502, probably never having received the money due for his sale of the title to Charles VIII. In his will, he granted his titles to Ferdinand II of Aragon and Isabella I of Castile, who never used them.
Although nothing ultimately came of the sale of the title, Charles had legitimately entertained the idea of attempting to conquer Constantinople from Sultan Bayezid II. Hopes for a French invasion of the Ottoman Empire ended when Charles died in 1498. Though Andreas once more claimed his titles after 1498, the French kings after Charles VIII – Louis XII, Francis I, Henry II and Francis II – also continued to use imperial titles and honors. Not until Charles IX in 1566 did the imperial claim come to an eventual end through the rules of extinctive prescription as a direct result of desuetude, or lack of use. Charles IX wrote that the imperial Byzantine title "is not more eminent than that of a king, which sounds better and sweeter".
Andreas remained an influential figure in Rome until his death. On 11 March 1501, Andreas prominently partook in the ceremonial entry of an ambassador from Lithuania into the city.
Andreas eventually died poor in Rome at some point in June 1502. In his will, written on 7 April that same year, he once more gave away his claim to the imperial title, this time to Ferdinand II of Aragon and Isabella I of Castile (neither of whom used the titles).
The question about how much claim Thomas and Andreas Palaiologos had to the imperial crown, and whether Andreas could sell or give away or will whatever hereditary claim he had, is hard to answer.
Theoretically the Roman Emperor position was always legally elective. but even from the beginning the emperors had their sons or other heirs designed as their successors or co rulers. So if Thomas and Andreas Palaiologos had hereditary rights to the imperial throne, did they have the right to sell those rights, or were those rights strictly hereditary, to be passed down to their descendants and not be alienated?
In posts 115 and 116 on page 12 of: https://historum.com/threads/heirs-of-the-byzantine-empire.121359/page-12
The heirs of two nieces of Andreas Palaiologos are traced down to the present.
I can't get that link to work. Go to Historum.com and European History and the thread "heirs of the "Byzantine" Empire?"
They are the senior genealogical heirs of the Palaiologos dynasty and its claim to the eastern Roman or "Byzantine" Empire.
Of course there are many other possible genealogical claims to the eastern Roman throne.