May I add two pennies to this discussion.
I would suggest that the two best places to seek long lived Roman family names are Ravenna (which remained under imperial control, even as the Western Empire collapsed), and Venice, which was never sacked by the Invaders (or anyone else). The original population of Venice consisted of refugees from Roman cities near Venice such as Padua, Aquileia, Treviso, Altino and Concordia (modern Portogruaro) and from the undefended countryside, who were fleeing the successive waves of Germanic and Hun invasions. I met a Venetian once who claimed that many families in the city could claim ancient lineages (including back to late roman times).
The historian Francesco da Mosto claims a humble but traceable family tie back to Venice in the fifth century - he has stated that many other families have kept the family tree tradition of the Roman times and can give similar family accounts.
Therefore, given that many noble families will have been amongst the refugees to the safety of this 'Roman' city in the 5th century (they after all could pay for the sea transport), then logically they will also have kept the family tradition of listing ancestors ....
Hopefully, someone will one day persuade these families to make public the family records and let the world know of the long histories that some them can claim.