According to the historical records of the Cornaro / Cornèr family of Venice, they have their ancestral ground from gens Cornelia, via the city of Rimini.
Here are links for Wikipedia (Italian version is more informative) and The Art of Living Long from Louis Cornaro, William Temple
the family can derive themselves back into Middle ages
so Cornaro / Cornèr family could be good candidate to an old Roman family surviving the Dark ages after Roman Empire.
history.stackexchange.com/users/11544/Lando-Cornèr
The wikipedia page on gens notes that Although both the concept of the gens and of the patriciate survived well into imperial times, both gradually lost most of their significance.
In the final centuries of the Western Empire, patricius was used primarily as an individual title, rather than a class to which an entire family belonged.
The gens originally held a governance function, then that governance function was absorbed into Roman governance.
After the Imperial period, the functions were no longer relevant.
There was no reason to maintain the lineage, and the "notable families" bred back into the general populace.
With respect to Latin surnames, I don't have any data on the distribution of surnames, so I don't know if Latin surnames are under-represented.
If I were to hazard a guess, I suspect that after the fall of Rome, the prestige of a Latin surname vanished, and I would expect that today Latin surnames are probably similar to other ethnic surnames in distribution - but that is merely a hypothesis and I don't have data to test it.
Update: Even when Rome controlled Europe, only a minority of the population would have had a nomen .
I cannot cite a source, but I believe that the majority of nomen were from the Urban Tribes, and therefore would not have been "notable".
The governor of a province and his staff probably had patrician nomen; the inhabitants of the province probably did not.
Again, a hypothesis, but I would be surprised to find surnames derived from nomen anywhere other than Italy, Byzantium and possibly Spain/Portugal.
history.stackexchange.com/users/1401/Mark C. Wallace
The main issue with sustaining a 'line' is that without modern medicine, a decent fraction of marriages will not result in children.
This ends that gens right there.
If you have children, they may die before reaching the age to marry.
In ancient times, child mortality was high, something in the 30-50 percent range.
Wealthy families might do a little better, but not a lot.
Having a taste for large families helps the odds.
However, rich Romans in the Republic usually restricted themselves to 1 or 2 sons, in order to not divide the estate so the son could have enough wealth for politics.
This put their gens in the "risky" category.
An added risk for powerful, influential families was proscriptions and political murder during the Civil Wars and Imperial age under paranoid emperors.
Entire families could and were wiped out that way.
Roman families didn't last long.
The Julius Caesares died with Caesar.
His adopted nephew's line ended with him.
The last known person with even a remote link to Augustus by blood was the Emperor Nero.
history.stackexchange.com/users/3116/Oldcat
Basically, they petered out and were replaced by new families who in some instances gratuitously grafted old illustrious names to theirs.
This process took place a number of times.
For example, according to one estimate , by 69 CE only 2% of the senators had republican patrician ancestry.
And that was before the large influx of provincials into the aristocracy.
Another thing to bear in mind is that under Roman law a new citizen (a freedman or a foreigner) took the nomen of the Roman who had sponsored his enfranchisement.
So, for example, practically all Gallo-Romans were named Julius Somethingus.
And finally, modern surnames have practically nothing to do with all this.
history.stackexchange.com/users/1569/Felix Goldberg
Over the centuries the original Latin gens were diluted.
Even in late Roman times such names were not common.
In many cases Latin families married into the families of leaders of barbarians and their name was lost as a surname.
Latin names do tend to persist as first names, however, indicating the continuation of their cultural heritage.
For example, the names Julius / Julia and Cornelius / Cornelia are Latin names common in everyday families.
Latin surnames have survived sporadically.
For example, there is one guy in the United States with the last name of Chrysogonus , one of the Cornelian gens.
It is possible that many old Latin names exist in mutated form.
For example, the common upper class Italian name, Schiapelli , may be derived from Scipio , another Cornelian gens.
Due to the use of different dialects and languages in Italy old names have in some cases undergone considerable alterations.
history.stackexchange.com/users/4346/Tyler Durden
Descendants of those families are still here under different names.
I think everyone will agree that there was a very big cultural shift in Europe during the decline and then fall of the western empire, so much of a shift that Latin died as spoken language.
With the cultural stance switching to the local cultures, a powerful family would attain local titles and create a new regime based on local rule.
As in England, France and the H.R.E., people became known by their jobs or by the places they lived or governed.
The wealthy heads of the great houses, I'm sure, did what they could to maintain their status and as such became big fat targets for every warlord and chieftain looking for loot or to legitimize their own seat.
War is expensive, children at the time not so much so many children were wed into local houses to ensure peace.
So to finally get to the point, many prominent houses of today like the house of Hapsburg-Lothringen do descend from Roman Patricians, if Vatican records can be believed.
history.stackexchange.com/users/27576/Eric Lund
My understanding is that some of the prominent Western Roman families that were possible lineal descendants of these families migrated to Constantinople and the Eastern Roman Empire as the Western Roman Empire fell.
Others stayed behind in other areas of Italy.
The ones who migrated probably intermarried, and were integrated into the elite families of the Eastern Roman Empire aka Byzantine Empire.
After the fall of Constantinople the surviving refugees fled to Italy, and others stayed behind as the Phanariotes under the Ottoman Empire
history.stackexchange.com/users/27586/Seanchaí
What happened to the numerous (patrician) families of Rome?  Like the Gens Julia for example.  There are no ancestors today leading back to them like for example the famous Medici family from renaissance Italy. I can't think of any Roman "last names" still in use.  What happened?
What happened to all the notable Roman families?
history.stackexchange.com
<ancient-history><ancient-rome><names>
history.stackexchange.com/questions/17508
history.stackexchange.com/users/8698/Twan
Descendants of the old families are still around with other names. Latin surnames probably fell out of grace after the fall of the empire.  There is no clear answer to this question, however, it is possible that the descendents of Roman noble families are still around, albeit under different names. Other theories speculated that they migrated or just petered out altogether after the fall of the empire.
It has been suggested that descendents of prominent Roman noble families are still around today but under different names. Some of the Roman noble families may well have migrated after the fall of the empire. These families gradually petered out, particularly after the fall of the empire.