About 30 Minutes into the film when Rocky's son is walking home from school with a girl, she introduces herself and than he proceeds to tell her that his name is "Robert." But in earlier Rocky's and in the closing credits, he is referred to as "Rocky Jr."
regarding the naming Rocky's son Robert, this makes sense if you've read the novelisation of Rocky 3. Rocky remembers meeting his trainer Mickey, introducing himself as "Robert, but my friends call me Rocky ,as in Rocky Marciano" It is never made clear in the films though.
Rockys son is never actually called anything in any of the films that he is in. He is only reffered to as Rocky Junior in the cast listings. If an actor or actress is given a name they must be payed more. Maybe that is why he never had a "name" before.