The name of a child, when mother and father disagree... Who, by law gets the final say?

The mother.

Since paternity can be questioned, I will vote for the mother.
But could not tell you the law.
Will tell you if two parents can’t come to terms on a kid’s name, they might as well file the divorce papers at the same time they file the birth certificate.

Law isn’t a factor.

Naming a kid is supposed to a joint operation by parents who are going to raise the kid. Like @josie says, if they can’t even manage to agree on the kid’s name, what does that say about the marriage?

The parents never married. Couple split during pregnancy.

Everyone are friends, even though the mother married another man and birthed two more children by him. They are all very good friends and the younger children treat the original father like an uncle.

The legal answer is “whoever signs the birth certificate.” There isn’t a specific statute addressing this or a huge backlog of case law. It’s just how the process goes: (1) baby is born, (2) birth certificate is created based on input the nurse receives from the parents, (3) birth certificate becomes legal when reviewed and signed by one of the parents. So if one parent or the other isn’t present for the signing, the one who is there could legally make any last minute changes to the name that they want.

There is a process for changing errors after the fact, but it requires a judge to sign off on the new information. If the parents come to some sort of agreement in the meantime, this process usually goes pretty smoothly. But if their disagreement is still intractable by the time they get to court, then the judge has to decide which parent has the better case. I’ve never seen one of these cases in person, but I imagine it involves a lot of sighs and rolling of eyes on the part of the judge.

Not “law”, but practicality, it is usually the mother that is asked by the hospital what name to put on the birth certificate. That includes last name.

Giving birth is automatically a determination for guardianship, except in cases where a mother is already determined unfit. Because of that, she is the one given the responsibility of naming the child, and for that matter, designating the father, at least until paternity is proven.
There are, in fact, many many birth certificates out there with the wrong man named as father. It remains that way unless legally contested and proven.

Most of the times couples reach a consensus. The only man I know who got away with naming his daughter is Robin Williams!

So, unless you can agree, or you pull something like, giving up your life in exchange of a kick-ass name for your kid, the mom will name the kid…

In the end, he or she can change the name to whatever they want once they are old enough…

If they split during the pregnancy then I would say the mother gets full “rights”.