From the link I posted about about Idaho:
If a name (for the baby) is not provided within 10 days, the baby will be given the mother's last name. The worksheet will then be forwarded to the State of Idaho Bureau of Vital Statistics.
Legally, you can choose any first, middle or last name for your baby
If she wants the baby to have her last name all she has to do is not name the baby, or give the baby her surname.
I imagine there are birth certificates issued that don't list a father -- like if the mother doesn't know or is a victim of incest or rape.
The father is denying the baby is his. Maybe it isn't. Why should he be listed on the birth certificate until it is proven that he is?
She asked a simple question: Do I have to give the baby the father's surname and will the surname have an effect on custody.
The answer is no and no.
You're trying to turn her question into something is isn't.