In 27 BC, the Senate gave Gaius Julius Caesar the title Augustus. I am assuming that as the senate gave him that title, the motion must have been proposed by someone at the senate floor. Do we know, reliably, who proposed that motion? Do we know how many people opposed it and supported it and how many (if any) senators abstained from voting?
My answer really focuses on the why as much as the who, because the reality may have been that in many ways he gave himself the title.
Octavian returned from Egypt with a wealth of treasure and a serious wealth of power. He was respected by his legions, of which he was commander of all sixty of them. Upon his return his fellow Romans believed they had seen the end of war and strife, and they hailed him as the Prince of Peace and benefactor of mankind. Octavian understood the importance of ensuring the people of Rome that he was benevolent. He certainly did not want to meet a similar demise to that of his great-uncle Gaius Julius Caesar (adopted father). So began a massive effort in religious renewal. Octavian restored 82 temples in one year alone. Also, there were grand new buildings like the Theatre of Apollo, and the Horologium. Octavian's right hand man Agrippa, too, embarked on several major building projects. Among these were the Pantheon, later rebuilt by Hadrian. Agrippa also repaired the city's water system and added two new aquaeducts, the Aqua Julia and the Aqua Virgo (Octavian famously quoted "I found rome of clay; I leave it to you of Marble") Rome was happy and Octavian had marble and possibly more importantly religion. Religion could be manipulated to suit the ambitions of Octavian and those he had given power to. Before we get to "those he had given power to" lets finish the topic of religion. Octavian recognized his prominence and equally recognized that without some shrewd decision making he had nowhere to go but down in regards to said prominence. His efforts to renew religious vigor throughout Rome would grant him a buffer during bad times and though one could assume that during good times he wouldn't receive the glory; they would be mostly wrong. Back to "those he had given power". The first form of "given" power would be the priesthood. Restoring temples meant restoring priests, of which Octavian had both subtle and secure power over. The second form and most noticeable to those with nobility was the purge and restoration of the senate. In 27 BC Octavian is famous for transferring power back to the Senate, but what really happened? The following two snippets do a great job of explaining this.
So in 29 BC two years prior to his "benevolent" action of giving up the power he ensured that he indeed would not really be giving up anything, well at least not in terms of power (He "had" religion and he had the Senate). It should also be noted that 29 BC was not the only time he reduced the number of Senators.
If you absolutely had to break this down into one person other than Octavian himself who would have been responsible for the title of Augustus being handed out, I would suggest that person was Marcus Vipsanius Agrippa. Agrippa was "his right hand man" but possibly even more important he was Octavian's highest military aid, closest consul, son in law, and soon after their return to Rome, a member of the Senate. That being said, Agrippa was just one of about 800 members in the senate at that time.
According to Cassius Dio:
What I would interpret from this is that while a number of people may have made suggestions on Octavian's new title, he was the one who really decided upon it and the Senate vote would have been pretty much a formality.
Cassius Dio does not record how the Senate voted on the matter, although it would be odd for there to have been any voices of dissent when they were not present on the more significant matter of granting him his powers. There is also no record of who suggested the name in the first place or who put the motion up in the Senate. I would consider Agrippa to be an unlikely candidate for the latter given that Octavian would not have lacked sycophants who could have done so instead and having his closest ally make the proposal would not have been in keeping with his habit of making honours look as though they were voluntarily bestowed by a grateful Senate rather than actively sought.