I am also a single mother and was concerned about my son having his father's last name (his father never came to the hospital even when our son was in NICU).
During our negotiation hearing, the hearing officer asked if my son's last name should be mine or his father's. I told the hearing officer it should be my name. His father didn't try to fight me (I'm glad because I cringe every time I see his name in his personal training ads). The hearing officer didn't press the issue because by then she was sick and tired of my son's father.
My cousin has a different experience. During her child support court hearing, her son's father didn't show up. One of the things that did come up was the child's last name. According to my cousin, the judge told her that her son's last name would have to be the same as the father's (it already was). That sounded strange to me, but I may have missed some details, so take that with a grain of salt.
I think it may depend on the judge or hearing officer.