My son lives in PA and the mom lives in NH. He wants to spend time with his child and the mother won't let him. There are no legal papers between them asar as custody.

Then he needs to file for child visits though the court. Only the court can order him specific visit times

Is he on the birth certificate? Also why hasn't he filed anything with the courts?

It is NOT legal until he has established himself as the legal father. Even once that happens, unless there is court ordered visitation, then he cannot go against the mother's wishes without court backing. So tell him to get a lawyer specializing in Family Law and start filings.