What if the step-father of a child wants to adopt the child of his spouse. There is no fathers name on the birth certificate.

Then you should retain a Family Law attorney that can help you process the adoption, so that he can go through all of the legalities of the process.

Having no name on the birth certificate actually makes it more complicated, as strange as that sounds.

A court will require going through a process to contact the birth father and establish paternity so that they can then have the father sign rights over to clear the way for your adoption.

Just has it right - get a lawyer. Each state is going to be different as to how this needs to proceed and a local lawyer will be able to best guide you.

Yes, not having a name makes it a lot harder, They will have to make an effort to determine who the father is, prove it with DNA ( if not already done) and get this person to sign over their rights for the adoption.