Not sure how strict their fathers were on them at home and these are all obvious names, but Tommy Gambino, Alphonse Persico, Michael Franzese, all had legendary mobsters as fathers and had varying degrees of success in the mob themselves.

I've read Barney Bellomo's father was a soldier with the Genovese.

Salvie Testa and Joey Merlino in Philly follwed their fathers into the life.

Joe Colombo's father was a soldier for the Profaci's and was murdered.

Maybe I'm wrong but it seems to me the Profaci/Colombo family has the most father/sons as members than other families.