The specific process is going to depend on the state issuing the birth certificate as well as whether the mother’s agreeable with having the father added to the birth certificate (won’t prevent it, but it’s a longer process).

If there’s no objection on the mother’s part, it’s usually as simple as your son filing a notarized declaration of paternity with the state and paying a filing fee to have the birth certificate amended.

If it’s contested (which I guess it may be if you have had a paternity test conducted), it’ll likely require that a paternity suit be filed to establish paternity.

Of course, if this is a situation where the mother’s not wanting him to be involved with the child, establishing paternity is possibly only the start of a drawn out fight to establish paternal rights, especially given his incarceration.

You can act as an agent for him and go to the county court house with the DNA result and request that his name be added if they refuse then you will need a lawyer.~