Which Italian families are you thinking of? The Italian states had a complicated, often contentious relationship with the Byzantine Empire. Consequentially, dynasties tracing their origins to the Byzantine Empire aren't very numerous. Rather, Charlemagne is the common ancestor everyone is eager to draw a line to. However, I can't stress enough than many noble families took a lot of liberties when drawing up their family trees, and prestige took precedence over facts. For example, according to the self-published history of the Medici family, when Charlemagne was "liberating" Italy from the Lombards, Averardo, an illegitimate son of the Emperor, valorously defeated a giant called Mugello in a battle somewhere northeast of Florence. Averardo and his descendants would reside in "Mugello's Region" from then on.

Noble families all over Italy claim descent from Charlemagne, and not all were sovereign rulers: for example, the earliest ancestor of the Carandini family (who were important vassals of the Dukes of Modena) went by the name Something-Something de Risi (we don't know his first name) who was a German soldier who accompanied Barbarossa to Italy in the late 12th century. Mr. de Risi was probably a distinguished knight that claimed descent from Charlemagne, although we have no way of verifying if it's true. Barbarossa did chose to reward Mr. de Risi with a bit of hand-waving though, letting him stick the imperial eagle on his coat of arms. Certainly, with Milan nearly demolished and its citizens scattered all over Lombardy, Barbarossa wanted to repopulate the city with people loyal to him, and appeasing Risi's genealogical demands would certainly be a means to convince his knight to stay behind in Italy.

Roman descent was also popular, for example; the Visconti would have us believe they were Romano-Celtic patricians living on the shores of Lake Maggiore since the aftermath of the Trojan War. However don't think the sources are airtight, 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 history of the Visconti was written in the 14th century by the Milanese chronicler Galvano Fiamma, who spends a great deal of time on the Visconti's great lineage 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.

Examples go on: the House of Este, which ruled Ferrara and Modena (and who are related to almost every other Royal house in Europe) are themselves are a Junior Branch of the House of Welf, following the 11th-century union of the Elder House of Welf (who were themselves a junior branch of Burgundian Royalty) with the House of Obertenghi, who were Lombard Nobility claiming descent from the Roman Gens Attia.

Like descent from Charlemagne, Roman descent was also popular among non-dynastic families, especially in the Republic of Venice. Dandolo family, for example, traces their lineage back to the Roman gens Ursia, specifically from Orso Ipato and his son Teodado Ipato, who were Byzantine Military officers when the Northeastern Italian lagoons were a Byzantine protectorate. Although the authorized Dandolo family tree finds a way to work around it, I have to point out that there is a two-hundred year gap in the paper trail between Teodado Ipato and the next Dandolo, one Domenico; a Venetian merchant active in Costantinople unremarkable apart from having somehow smuggled the remains of St. Tarasius from Costantinople to the Basilica of St. Zaccaria in Venice.

The Badoer family of Venice also traces their origins to the Gens Ursia, and in the early tenth century there is indeed record of one Ursus Badovarius applying for a subsidy when raving Maygar hordes devastated his land somewhere near Altino at the edge of the northern lagoon. That area between the mouth of the river Sile in the northern lagoon and the fortified town of Eraclea was one of the primary settlement areas for Byzantine soldiers, so I suppose the claim is realistic, but really we're taking the Badoer on their word here. The examples among Venetian families go on; were the Contarini descended from the Roman Aureli Cotta as they claim? We don't know, the first we hear of them is a jurist in the early ninth century underwriting a will. And the Giustiniani trace their heritage to Emperor Justinian! We have spotty documentation of a Byzantine family called Giustiniani in Istria that arrives in Venice sometime before the eleventh century, but that's it.

I can think of one very specific Italian Noble Family that can document their ancestry to the Byzantine Empire: the Marquis of Monferrat, who descend from a cadet branch of the Byzantine house of Palaeologus. The Marquis John of the ancient Aleramid house died without heirs in 1305, and the duchy was inherited by his sister, who just so happened to be married to the Byzantine Emperor Andronikos. The March passed to her second son, Theodore, who promptly set sail for Italy, married Argentina Spinola (the daughter of the powerful ruler of Genoa, Opizzino Spinola) and with Genoese support was able to extract the Monferrat from Philip of Acaia, who ruled over the Piemontese domains of the County of Savoy as vassal of his uncle Amadeus V and had seized the Monferrat as soon as the Marquis John had dropped dead.

The Monferrat continued to be ruled by the Paleologus until the Marquis Giovanni Giorgio died without male heirs in 1533 and the March was inherited by his son-in-law the Duke of Mantua Federico II of the House of Gonzaga. The main line of the Gonzaga went extinct in 1708 and unfortunately the only branch that still exists to this day predates the union of the main branch and the Paleologus.