How do I go about adding my son's father's name to the birth certificate?

Well what you have to do is go to your local hospital and speak to person from birth records and tell her you a need the paper work for the father acknolegdement and the you will both sign it and get it sent out its very simple

I am sorry but in general after the birth certificate is issued, the hospital can no longer do anything. You will have to petition the court to make the change, once the original birth certificate has been issued.

NO I know that because I just did that for my son I didn't have to go through any court petition.. they can give you that paper work at a local hospital