Quote Quoting Mr. Knowitall
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Your child's name has no relevance to your obligation to support your child.
I agree with this response.

However, I will also add that if its really mom's boyfriend's last name that the child wants to use, rather than mom's husband's, I would strongly recommend saying no for the child's sake. If mom and the boyfriend do not stay together permanently, the last thing the child is going to want is to have her boyfriend's last name.