Can my husband add himself to his sons birth certificate?

There must be court orders that give custody to your husband.  Review the existing court orders to see if they establish your husband as father.  If they do, then he should be able to apply to the Texas State Department of Health- Burequ of Vital Statistics to add him as the father to the birth certificate.  If the current orders do not establish him as father, then he will need to file something to ask the court to establish him as the father and to expressly order that he be added to birth certificate as father.   The mother may need to participate or cooperate with the application to the state for the birth certificate.  She will have to be served with the law suit if a law suit is necessary.  Start from the address for her that is listed in the most recent court order.