If you legally acknowledge paternity by including the father on your child's birth certificate, then your child will be eligible to receive Social Security death benefits. Plus, the idea that including him on the birth certificate makes it easier for him to get custody is a misconception.In reality, your baby's father can formally request custody or visitation at any time â whether he's on the birth certificate or not. All he has to do is file a request with your local family court.irst, you should know that legally including the father's name on a state-issued birth certificate requires his participation. So if he happens to be unavailable â because you don't know who he is or cannot locate him â it's a non-issue.