My sons father lives in another state and I need him to sign his birth certifact but he is incarcerated how do I go about that

Ask the hospital their rules, it could be sent certified mail and signed with notorary at prison. If the hospital will do this.

Have someone given his power of attorney to sign for him.

Just some thoughts, but there are tons of fathers in prison, so most liekly the hosptial has a procedure for it