I'm agreeing w/ those about the child having the mothers last name. I was not married when I had my first child 11 years ago, I put the father on the birth certificate, but b/c he was not there and did not fill out the paperwork - her birth certificate was blank under "Father". I gave her my last name, b/c she was a part of my family - I wanted her to have the last name of the family she was being raised by.
In my opinion, to children the last name can feel important. A few years later, I married a man who was not my daughters biological father. We had 3 boys and they all had my husbands last name, as did I when I married him. My dd always knew she was part of the family, my husband has raised her since she was 18 months old, as we tell the boys "your sister was here first" - and in her concious memory, my husband has always been daddy to her. They are "thick as theives" as they say. But when we'd get gifts that said "To the _________ Family" - she'd often feel sort of sad, or left out b/c she had a different last name. She hated when she was in sports and her last name was different than the rest of the family. While she is very secure in her realtionship in our family, she still felt left out. It took us some time to come up w/ the money, but in 2005 my husband adopted her and her last name was changed to his. It was amazing how much of a difference that meant to her! She is so proud of her last name now! She is so proud to have the same last name as her siblings and her daddy.
I realize it's just a name, and not all kids are that way, but personally, I'd hate having the last name of parent I did not know (if that ends up being the case in your stepdaughters relationship).
If he does not fill out paperwork, he will not be listed as the father on the birth certificate. She could give the child the father's last name, but if there is not a good relationship, it's my personal opinion it would be better to give the child her last name so the child always feels that "family" connection.
The last name has no effect on child support. The child's last name could be Santa Claus and it wouldn't matter. If she goes after him for child support (b/c he is unwilling or disappears) - then he may protest paternity, and he will be required to take a paternity test, at his expense (he'll have to reimburse the state). I would just make sure she knows his social security number if you predict a negative outcome about their relationship.
Good luck to her!