Legal Question in Family Law in Texas

You'll need to go through court and do a formal name change. After that, once the court has entered an order changing the child's name, you file an application with the Bureau of Vital Statistics (in Austin) for an amended birth certificate.

If Dad relinquished his parental rights, and there is a Court order terminating his parental rights, the order itself should have a provision in it that changes the child's name.