you need to file with the courts an action to change his name. The father will get to respond to the action and either agree or contest it. If he agrees, no problem. If he contests it, the judge gets final say.

here is the website from Franklin county but it should be very similar if not identical across the state:

http://www.franklincountyohio.gov/pr...How_Do_I_Apply
You go to the courthouse in the county of the child's legal residence