How did ancient Roman family names survive to current day use?

Just to add to what others have said, Burckhardt has some very interesting comments on this topic in his Civilisation of the Renaissance in Italy under the chapter 'The revival of Antiquity'. A short extract:

When the enthusiasm for the ancient world was greater than for the saints, it was simple and natural enough that noble families called their sons Agamemnon, Tydeus, and Achilles, and that a painter named his son Apelles and his daughter Minerva. Nor will it appear unreasonable that, instead of a family name, which people were often glad to get rid of, a well-sounding ancient name was chosen...Thus Giovanni was turned into Jovianus or Janus, Pietro to Petreius or Pierius, Antonio to Aonius, Sannazaro to Syncerus, Luca Grasso to Lucius Crassus. Ariosto, who speaks with such derision of all this, lived to see children called after his own heroes and heroines. Wikipedia:The Civilization of the Renaissance in Italy

People in the Roman world discontinued using complete Roman names over a period of centuries in late antiquity and the early middle ages. By about 500 or 600 almost everyone would have just used their personal name, without any clan or family name.

I note that the eastern section of the Roman Empire continued for centuries after the western half fell, but Greek replaced Latin more and more. Aristocratic families there started using Greek surnames at least as early as before the year 900. I think that Emperor Romanus I Lekapenos (r. 920-44) was the first emperor to have a surname, and that he was the first in his family to use it.

Surnames developed more or less independently in western Europe. I have the impression that in rural villages tax collectors were responsible for surnames. Since several villagers would have the same personal name, the tax collectors wrote descriptors after the names in their list, descriptors which were used thoughout a person's life and eventually became hereditary.

Nobles in England began to use surnames in the middle ages. The first Scottish kings to have surnames were John Balliol (r.1292-96) and Robert I the Bruce (1306-29). I believe that Edward IV (r.1461-1483) was the first king of England to use Plantagent as his surname, and Henry VII Tudor was the first noble with a surname to become king of England. And even the present royal family doesn't have a real family name.

In larger towns and cities in western Europe, there would be many more men of the same personal name than in a village, so telling them apart was a bigger problem, and so surnames may have begun earlier than in villages, and used by aristocratic families.

Apparently all the Doges of Venice had surnames since 742. So apparently the use of surnames in the Italian language was common in Italian cities since quite early in the middle ages.

Wikipedia.org:List_of_Doges_of_Venice

And during the Renaissance it became common for educated people to Latinize their names, which included surnames. So probably a lot of people adopted ancient Roman clan and family names.

I just want to address a few points that might help situate how to view these names. First, Cicero wasn't a last name, and Roman names did not follow the post-antique first-middle-last naming structure.

Cicero's surname (Latin: nomen) was Tullius. His personal name (Latin: praenomen) was Marcus. His family "branch" name (Latin: cognomen) was Cicero. His daughter's name was merely Tullia; his son was also Marcus Tullius Cicero. But if that son had been adopted by e.g. Marcus Livius Drusus, his name would have changed to Marcus Livius Tullianus, with the latter name indicating his former family name. Originally, the name (cognomen) Cicero was actually just a descriptor, but famous persons often bequeathed that cognomen to their descendants.

Moreover, as sds mentions in the comments, last names fell out of use and were only revived in the late Medieval period. Plenty of documents during the early Medieval era mention people with a single name only. See e.g. this list of Patriarchs at Grado. A second name is absent early on and only starts to appear consistently in the tenth century.

You can see the same with the list of popes. They (almost?) all have a single name until the 11th century, when John XVII was born Giovanni Sicco. Before that you get a couple of people who were often labeled by their hometown as well, but not very many.

So no, the last names did not survive the Middle Ages. Maybe you can find someone with an unbroken chain of names going back that far, but they'd likely be royalty. Otherwise, the Roman names are just tacked on later.

I believe a complete prosopography of Medieval names is lacking and so the statistics you're looking for will be hard to come by, but people did adopt ancient names. How this particular name (Cicero) came to be adopted is anyone's guess.