Regardless of the name given to the child, the father has the same amount of rights. You need to see an attorney to set up child support and see if the father can be asked to help with the medical costs. If he's not interested in the child, he can sign off on his parental rights, but he still is required to help support the child.

I think by worried sick's account it all depends on which country you're living in, I'm in the UK and my son said that he did have more rights with his name on the birth certificate, so - you can pick between us!!