My son is 5 my x boyfriend got arrested when I was 6 months pregnet.I gave my son his last name because we were supposed to be married when he was released that day came and I never heard from him.My son will be starting school and I want him to have my last name also there is no father listed on birth certificate I live in nyc what can I do

By not listing the father you have made this harder. You have to go to court to effect the name change. The court will require that you identify the father and attempt to get his permission for the change.