Hi OP. You wanted a male perspective. I'm a single dad, with 50/50 residence of my 14 and 12 year old daughters.
My perspective is that neither your son nor the mother are coming out of this well. Let's start with the fact that the mother has misrepresented the parentage of the child on the birth cert, and (giving her the benefit of the doubt, that she didn't know this to be a lie when she did it) is now refusing to fix the situation. That's very shoddy behaviour - every child should have the right to know who they are, and where they come from. Deliberately placing parental responsibility on a man who isn't the father, and denying it to the real father, is bloody awful behaviour. Unfortunately, it's standard advice on MN for women to refuse to put the father on the birth certificate, in a bid to deny him parental responsibility. It's crappy human behaviour, and (in the absence of a genuine safeguarding risk) just a power play from a controlling breed of women.
So, she is in the wrong on that front. But, as others have said, if she won't voluntarily fix it, then your son's only recourse is to the courts. If he isn't willing to do that, then he won't get parental responsibility.
Now, let's deal with your son. He is seeing the child every weekend, so he is involved with his child. That's a good thing. The question of why he wants to change the birth cert is an important one - the child has now reached an age where they know their name, and it is bound up in their sense of who they are. That has been distorted by the poor behaviour of the mother, but nonetheless it is established now. Courts are very reluctant to change the name of a child, and for good reason. In my view, he would be wrong to push for a change in the child's name at this point. Doing so would be more about control, than about the best interests of the child - and that's not cool. So he would be right to go to court to get parental responsibility, but wrong to push for a change of the child's name.
People have focussed on money for good reason. I'm going to assume that your son isn't seriously disabled or something, that prevents him from working. Stepping up and being a dad is about more than being recognised legally as the father, and spending a few hours together at the weekend. It's also about the hard graft, and providing for his child is part of that. Those of us who are involved, hands-on dads who day-in and day-out put our kids first, and work hard to give them the best life possible, have little to no time for guys who try to shirk those responsibilities. They're a big part of the problem - they contribute to the sense that some parts of society have that dads are somehow second class parents. That's bollocks, but guys like your son don't exactly help the rest of us who are doing the right thing, to overcome that perception.
So, your son would be right to seek parental responsibility. But being a dad - a proper one - is about way more than that, and he needs to step up fully, and do right by his child. Until he does that, the sympathy he gets from the rest of us will be very limited.