she can list the father as unknown.
now to receive child support, she has to establish paternity. dna testing, or the father would have to admit to paternity and sign the birth certificate.
when paternity is established a support order can be established.
when paternity is established the father can also petition to have his name added to the birth certificate - if the state handles these things they will petition to have it placed on the certificate. he can also petition for a name change in the child - if he wants. he can also petition for a form of custody and visitation.
when paternity is established - both parents have equal rights to the child. she would need to petition for custody to be established.