Where you are not married you would have to be adjudicated the father of the child to have standing to do anything. That being said, assuming the child is being properly cared for neither you nor your girlfriend should have anything to fear, she should just refuse to go with her mother. Absent a court order, your girlfriend's mother has no rights to see or control any aspects your child's life. If your girlfriend's mother is doing anything that is negatively impacting the health, safety and welfare of your child then court involvement could be sought. You could, without being adjudicated the father, obtain a no trespass order, but this would only keep her away from your home. If you fear the child could be physically harmed and have evidence to support this, then a restraining order could be sought on behalf of the child. Either you would need to be adjudicated the father or your girlfriend would have to seek this.
As Attorney Lukowiak pointed out, since you and your girlfriend are not married, the only person you can legally keep your girlfriend's mother away from is yourself. As your girlfriend is an adult, whether she would like to take legal action to keep her mother away, or simply wants to ignore her mother because she's legally entitled to do so is up to her. As your girlfriend is your child's only legal parent at this time, only she can do things on his behalf.
You might be able to seek an abuse prevention order on your own behalf, though the facts you have stated wouldn't support that. Another option is a harassment order, which doesn't require fear of physical harm but does require a showing that you are being harassed.
A more important thing for you and your girlfriend to consider might be getting a judgment from the Court that outlines both of your parenting rights. Doing this now, while the two of you can come to an agreement, can save a lot of energy/effort/money in the future. Your agreement could outline, for instance, that you are both legal parents, that you share legal custody (the right to make decisions), and that you currently reside together and therefore will not have any child support or parenting schedule for the time being. That way if, for instance, your girlfriend falls ill and you need to be able to enroll your son in school or take him to the doctor without her, you have the legal power to do so without needing to run into court in a state of emergency.
What my colleagues have advised you is correct:
1- Your girlfriend is the legal parent and guardian of the child since you are not married.
2-You do not have any standing until you get a court judgment that says you are the father and entitled to legal rights over the child (again, b/c you are not married to the mother)
3-You can't do much more than get a anti-harassment order to keep your gf's mother away from you; only she can keep her son away from her mother. bear in mind the gf's mother has to actually be harassing you. Interfering with your relationship with the child will not suffice.
If you want your rights as a father, file a paternity petition in the local Family court. If you and the mother are in agreement that this is what she wants done, it should be easier to get the order. If she opposes you then you'll know its not just the mother who is keeping the child away from you.
This answer is provided for informational purposes only and it is not intended as legal advice. Additionally, this answer does not create an attorney-client relationship. If you wish to obtain legal advice specific to your case, please consult with a local attorney