My grandson is 4 years old and my daughter would like to use her last name now that he is going to school. On the birth certificate is his fathers last name, but she would like to use her last name since the father is not in the picture, but does pay child support because he is in the navy.

A person can use any name they want as long as there is no intent to defraud. Your daughter will need to explain to the school when she register her son that she wants him to be known by her last name. Most likely the school will go along, but for anything that requires his legal name, they will have to use what's on the birth certificate.