I live in NYS. My 3yr old had the last name of her mother, but I have primary custody. The mother typically does not show up for custody herarings. I want my child to have my last name. What if the mother refuses?

You have to give the non-custodial parent notice of the name change and there has to be a "good" reason for the name change. If the other parent doesn't consent you can attempt an "informal" name change - but the other parent can stop that, too.

How to Change a Child's Name in New York | Legal Assistance