Well if you mean a specific name for businesses including "and sons" I couldn't tell you, however if you're talking about businesses being a kind of family affair...

...well the various republics of the renaissance, though most notably the merchant republics, had a kind of "patriarchy" (in the Roman sense, not the Feminist one) where businesses could and would be passed down through the family, with your children apprenticing under you at a fairly young age. This makes sense mostly because devotion to your family was (and still is) considered an incredibly important trait, and the relative speed of communication and isolation between different villages in that time meant it'd be pretty easy for someone to take your goods and run off with them.

It'd be much harder for children to try and steal from their parents and start a new life somewhere than "Meldrick" the local dung gatherer to agree to deliver your goods to another merchant's stall, only to run off and stay at his distant cousin's house with all your stuff. Plus names such as "Miller" or, to use an Assassin's Creed reference, "Auditore" have their root in professions, namely someone who runs a mill for "Miller" and an Auditor for "Auditore", the fact we past these names down through our family could hint towards people staying in their parent's professions.

There's also the matter of name recognition. It would probably be much simpler for the sons of, say, Accountants in Firenze to stay with their family and apprentice under their father, in part because when their father passes on they can still retain a kind of "brand recognition". People would say "Oh yes those accountants in Firenze, I knew their father, he never once cheated me."