You can change his name through the courts and they will issue it changed on the birth certificate. Did his father sign an acceptance of paternity?
http://www.uslegalforms.com/paternity/missouri-paternity-...
You should be able to handle this yourself with some leg work if his Dad is not fighting it. If he has not signed an acceptance of paternity you can't just put his name on the certificate and change your sons name. You will need to go through the courts and establish paternity.
My husband adopted our son when he was a year old, they added his name to the birth certificate and legally changed my sons name. We were issued a new birth certificate showing both changes.