As of right now I'm not on the birth certificate but after petitioning for paternity and custody/visitation rights would I be able to force a last name change since my son's mother didn't use my last name on his birth certificate? What are my rights in New York State?

A mother can name her child anything she wants to.

She doesn't have to give the child your last name. She doesn't even have to give the child HER last name if she doesn't use yours.

You'd have to prove, in court, why forcing her to change the child's last name would be in the CHILD'S benefit.