None at all. Actually, there is not a single solid case of continuity between noble families of the Middle Ages and Roman families; certainly, in areas of relatively smooth post-Roman transition (Gaul, Spain, Italy), most if not all noble families of the, say, 8th century must have had some Roman ancestors, since local nobles gradually mingled with Germanic incomers. The most serious attempt to document one of these connections was done with the Carolingians, which have been tentatively linked with a couple of important Gallo-Roman families. There is no doubt, however, that a considerable part of Charlemagne's ancestry was Frankish, and that even his possible Gallo-Roman ancestors had come to consider themselves as such by the 7th century. As for purely Roman (i.e. from the city of Rome) families, some of them may have retained their importance into the Ostrogothic period and later, but once again, there is absolutely no way to find a connection with more recent periods (actually, the obscurity of 7th century Italy makes it less likely than in Francia).

(as a side note, Odoacre did not conquer Rome, but rather Ravenna, the centre of power in the WRE at that time)