I have a baby girl named emma, the mother doesn't want her to have my last name. What rights do I have to get that changed? I pay child support...

If you were not married to the mother, you have little say about the child's last name, Did you ask for this in the custody agreements, but normally in court you will get vistiation, and you will have the proper amount of child support ruled on. But I doubt you can get the child's name changed.