This is a tough call. Under Texas law, without being on the birth certificate, not having an Acknowlegement of Paternity, Adjudicated by law, or being married to the mother, the father would not have any legal rights. That said, it would probably be good to have the father come to your state (jurisdiction over the child) to establish his paternity. Once that is done, then the Judge can order visitation, etc. Plus, your son could have a growing relationship with his biological father.
However, sometimes that can be difficult, especially if your child wants to see him. The legal process is long. That said, I would need more facts to really answer the question.
This is just an opinion, and is no way to take the place of an attorney that knows all the facts of the case. Therefore, I would seek such advice as soon as possible.
You do not have to be on the birth certificate to have rights. If the mother does not agree that he is the father, he just has to go to court to establish paternity-which can be done at any time- and she may end up stuck with the cost. You can agree to a visitation schedule that suits both parties or the court can be called upon to set one.
This answer is for general information purposes, is not legal advice and does not create an attorney-client relationship. If you wish to schedule a consultation regarding your particular case, please call (757)533-5400.
Sign up to receive a 5-part series of useful information and advice about child custody law.