How important is a last name ???

My friend is expecting a baby with her now ex BF. Since they weren't married nor was he attentive when she mentioned her pregnancy, she decided to give the baby her last name. He is the first guy she messed with since 2003 && He knows this but since she won't give the baby his last name, he's demanding a paternity test. He then told her good luck with the pregnancy and have a good delivery. They haven't spoke for the past two months and she was due last week. How important is the baby's last name when the parents aren't married ???