Most of the names which the families are knows was gave during the Valachi hearing using the name of the borgatas bosses in 1963:

Vito Genovese borgata
Carlo Gambino borgata
Tommy Lucchese borgata
Joseph Bonanno borgata
Joseph Colombo borgata

and so on for the other small families.

Out NY there was some families like Buffalo that was knows with a nickname as the Arm and the Patriarcas as The Office and Chicago as the Outfit and the Detroit Partnership.

Philadelphia Mob insted was called Bruno family and Bruno-Scarfo family and today the Merlino Organization.