You have to go to court to demonstrate that the name change is in the child's best interest. The question is who is listed as father on the birth certificate (and not subsequently officially disestablished). The two of you (mom and the "listed" father even if it he is not) can enter into the petition together. If no one is listed, then obviously just you.

The court in your county may have an affadavit that you can just fill in. I was able to find one for Davidson County online.