This is something I wish could somehow be implemented into Republics. The Patrician Family system in CK2 was pretty nice, makes me really enjoy playing CK2 as a minor family. Of course, you don't play as a 'character' in EU4, but it would be good if some kind of family system operated within the nation. Some families maintained control of republics for multiples of years, however at the moment, Medicis disappear after the first election and are never heard of again as generic, unheard of families take over. Too late for change in the republics for Common Sense, but perhaps next expansion, some fixed up mechanics for the Republics would be so good.
But yeah, just tank republic tradition since there's no other way to keep the Medici otherwise. And do it really fast. Or cheat.