The Roman Senate continued to meet throughout the first part of the sixth century, and even enjoyed a renaissance of sorts under the barbaric rulers beginning with Odoacer and especially under Theodoric the Great. The Gothic Wars, though, devastated Italy in the mid-sixth century, and Rome was no exception. In 536, it was recaptured by the Eastern Romans; a siege began shortly thereafter in March 537 and lasted for a year; sacked in 546 by Totila (during the siege of which, a famine haunted the city); captured again in 550 by Totila; and finally captured for good by the Eastern Roman General Narses in 552.
Nonetheless, the Roman Senate did survive all these events, albeit in diminished form. We know that they continued to meet in some fashion because they pleaded Emperor Tiberius II Constantine for help against the Lombards in 578, sending an envoy to Constantinople with 3,000 pounds of gold. (The Emperor returned the gold, saying there were no troops to spare and instead advised the Senate to spend it on using it to secure support from Lombard and Frankish rulers)
But by 593, Pope Gregory I wrote the following in his Homilies on Ezekial (essentially a reflection on the dire state of the world):
Where is the senate? Where are the people? The senate is vanished, the people have perished... Rome is empty and yet Rome is burning.
By 593, Rome of course was not empty, and was probably a city of some 30,000 - 50,000, making it still one of the largest cities in Europe at the time even though it was a shadow of its former self. So what he said should not be taken literally, but it's indicative of the decline of the Roman Senate.
The last time the Senate is mentioned though is in 603 in the Gregorian Register, in which it is noted as having acclaimed new statues of Emperor Phocas and Empress Leontia. But, it's referenced in the register as, "by the whole clergy and the senate." Moreover, the Pope ordered the statues to be moved to the chapel of the Imperial Palace on the Palatine. So it seems clear that in whatever form the Senate existed by 603, it was clearly no longer a significant body. After that time, there's no more references to the Senate, and we know that in 630, its meeting place (the Julia Curia) was converted to a church.
I think it's worth quickly noting though that the Eastern Roman Senate continued to meet in Constantinople for hundreds of years afterward. It was mostly a ceremonial body during this time, although it did have some influence (in 1197, they exempted Constantinople, and thus themselves, from a special tax that the Emperor had specifically convened them to approve).
As for the last person to hold the position, I have no idea. Boethius is probably the last well-known Roman Senator (don't take my word for that), but the Senate continued to function for decades after his death in 524. The last consul was Anicius Faustus Albinus Basilius in 541. After him, the title was added to the Imperial title until Emperor Leo VI got rid of it altogether in the late 9th century.
https://books.google.com/books?id=vQFBAQAAQBAJ&pg=PA3#v=onepage&q&f=false