If you have SOLE LEGAL custody of the child, then all you have to do is petition the court for a name change. If you have JOINT LEGAL custody, then you won't be able to change the name easily - you'll have to get the court to agree to do it even though the father has the right to refuse such a request. You'd have to prove he is missing, whereabouts unknown, and that you undertook lots of effort to locate him, plus publish notice of the proceedings, etc.
This answer is not to be considered a response to a specific legal issue in a specific jurisdiction - it is to be considered only a general response to a hypothetical scenario posed by the questioner. For specific legal advice, please consult with a licensed attorney in your jurisdiction.
In North Carolina there is a specific appellate case on point. I was involved in the writing of a brief for it back in 1999.
The case is In re Change of Name of Crawford to Crawford Trull, 134 N.C.App. 137. It says: "Neither the clerk of superior court nor the superior court judge erred by denying a petition to change the name of a minor where the parents were never married, the natural father's surname was given to the child on the birth certificate, and the mother sought to change the surname to her own over the father's objection. N.C.G.S. § 101-2 does not permit one parent to change the name of minor children without the consent of the other living parent and respondent here clearly fits an ordinary definition of “father” and “natural parent.”" That case further says that there is not a best interest analysis for a name change in such circumstance.
With what you have said it may be that your next step is to consider a termination of parental rights of the father.
LEGAL DISCLAIMER: SCOTT ALLEN IS LICENSED ONLY IN THE STATE OF NORTH CAROLINA. THIS RESPONSE DOES NOT CONSTITUTE LEGAL ADVICE AND DOES NOT CREATE AN ATTORNEY/ CLIENT RELATIONSHIP. RATHER, THE RESPONSE IS IN THE FORM OF LEGAL EDUCATION AND IS INTENDED TO PROVIDE GENERAL INFORMATION ABOUT THE MATTER IN QUESTION. ALTHOUGH A RESPONSE IS PROVIDED TO THE SPECIFIC QUESTION, THERE MAY BE OTHER FACTS AND LAW RELEVANT TO THE ISSUE THAT THE QUESTIONER HAS LEFT OUT AND WHICH WOULD MAKE THE REPLY UNSUITABLE. THEREFORE, THE QUESTIONER SHOULD NOT BASE ANY DECISION ON THE ANSWER, BUT SHOULD CONFER WITH AN ATTORNEY IN PERSON ABOUT THE SPECIFICS OF HIS OR HER CASE
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