Keep in mind that family names are usually assigned by law enforcement or the news media. Guys inside the family usually refer to their families not by the name given by the above, but just generically, as in "our borgata," or "I'm with Dom" (or whoever).
At that, before the Valachi hearings in '62, neither law enforcement nor the media referred to individual families by name. They just referred generically to "the Mafia," "the Mob," or "organized crime," as if it were a single entity. Nor did they pay much attention to the Mafia before Valachi. He was the one who revealed the details of structure, and the bosses.
Nor is it clear to me that Dons--even egotistical ones like Gotti--ever impose their names on families. Sometimes, if the media or law enforcement sense that the new boss has an outsized ego, they might assign a new name--that's probably how the Genovese family got its name. The Columbo family probably got its name from the high profile that Joe Columbo took with his Italian American Civil Rights League. I doubt that it was referred to as "the Profaci family" before that.
Now, you're thinking, "If outsized ego is the figure of merit, whey weren't the Gambinos renamed the Gottis?" Probably because Carlo himself was so widely regarded as the top Mafia guy in the country. Gotti, for all his high profile, was arrested and on trial so often that I think the media probably regarded him as "temporary."