Although the declining Roman empire left lots of room for non-patricians to make a name for themselves (especially in the army, more on that here, aren't self-plugs fun!) many old Roman families were alive and well (albeit often refreshed with new blood. As stated in the link above, the last emperor, Romulus Augustulus was himself half-roman, half "Barbarian").

Here are some anecdotes of interesting families I'm familiar with (adapted from an earlier answer):

The Visconti, lords of Milan from the 13th to the 15th century (and active in Lombard communal politics since the 10th century) claim to have been landholders in Massino on the shores of lake Garda since the land was granted to an ancestor in the days of Imperial Rome. They also claim descent through the Roman era from Eneas, the ancient Trojan hero (mind sources on this are questionable and propagandistic, many conveniently having appeared when a match was being made between Gian Galeazzo Visconti and Isabel of Valois, daughter of the King of France. The most complete chronicle was written in the 14th century by the Milanese chronicler Galvano Fiamma, who spends a great deal of time on the Visconti's great exploits in his Politia Novella, or modern history of Milan, today stored in the Ambrosian Library. The tone he takes when speaking of the Visconti is unsurprising given he was Duke Giovanni Visconti's personal chaplain).

Although the main Visconti line died out in the 15th century (and the throne passed to the Sforza, a cadet branch that later went extinct) a couple of cadet branches continued to exist in obscurity as small landholders in the Lombard countryside. Many are involved in business and real estate in North Italy up to this day, the most easily traceable being the line of the Visconti of Modone, descendant of Vercellino Visconti, younger brother of Matteo Visconti (1250-1322) who was the Duke of Milan from 1294 to his death (with an eight year-exile after the Della Torre seized back the lordship, but that's a story for another time). The family was clever with their allegiance even after the throne of Milan slipped from their family, and several members were modestly important diplomats and civil servants in the court of the Sforza Dukes. They themselves were elevated to the title of Dukes by Napoleon in the early 19th century. The family count among their number Lucchino Visconti (1906-1976) and his brother Erpirando Visconti (1932-1995), both successful film directors. For a while I thought that this was the last branch of the Visconti to go extinct, but I just checked and it turns out I was wrong; Erpirando does have descendants, and they live in a modest villa near Bologna. It appears they do not own any more land near Modone, but have significant wealth managed privately.

Another cool example are the Obertenghi, who descend from Boniface, a Bavarian ennobled with significant lands in Northern Italy by Charlemagne (he was granted nearly all of modern Tuscany, Emilia, Lombardy, Peidmont and Liguria). The Obertenghi saw their lands reduced to Lombardy and Emilia by the eleventh century, and finally split into three cadet branches by the twelfth century: the Estensi, Pallavicini, and Malaspina, all three mildly influential during the communal era and increasingly obscure when the communes were consolidated into the dozen Italian states of the renaissance. The Estensi, who became lords of Ferrara, are the most illustrious and are related to nearly every royal house in Europe. They were also the first to go extinct. The Malaspina, the Tuscan branch of the Obertenghi, became Marquis of Massa-Carrara, but had their holdings progressively reduced under the Florentine Republic and subsequent Grand Duchy of Tuscany. They split into various branches living along the Tuscan-Ligurian border and as far as I know all went extinct (although I don't know as much as I'd like on Tuscan history). The last branch are the ones who answer your question: the Pallavicini, already active in Emilia and Lombardy in the communal era, became powerful vassals of the Duke of Parma, but wisely allowed themselves to slip into obscurity during the Italian Wars. To this day there are Milanese and Emilian (based in the town of Zibellio) branches still in existence, both are active real estate developers in the Emilia-Romagna region.

I'd also like to mention the Dandolo, a family that gave the Republic of Venice four Doges. A branch of them is still extant today and inhabits one of their original palaces along the Grand Canal in Venice. The Dandolo family traces their lineage back to the Roman gens Ursia, who arrived in Venice in the eighth century from the Pentapolis (the last five Roman cities to be abandoned by the withdrawing Byzantine empire). However, I am not sure they still hold the deed to the other palaces previously owned by the family (I know of two; one is a hotel, the other holds Venice's city hall) or what sort of activities the descendants partake in.