If someone knows your birth name and date they can theoretically find any birth relative, they don't need your "permission". However, if they don't know it and you either don't know or won't tell, then they're not going to get anywhere I shouldn't think.
You can obtain a copy of your original birth certificate whether you know your name or not, but as far as I know they would have to commit fraud to get it, unless they know your birth name. I don't know if social services would be any help to a grandchild trying to trace birth family, they aren't much help to adoptees as a rule!
Try After Adoption for some help with this. The helpline is great. They should be able to advise whether or not your son could trace and help you both come to terms with the consequences of whatever he decides. They won't try and push you into tracing your birth family so don't be worried about that (though they are a great support if you do want to do that at any time).
Maybe stay off the forums though, they are more for "curious" adoptees.
It sounds to me like some fears have been stirred up by your son's girlfriend asking questions, which is absolutely fair enough.
You could get some support from AA to work through what it is that upsets you so much about it and your son could get some insight into why this line of questioning is unwelcome - the girlfriend sounds tactless at best.
I do understand you don't want to upset your mum, but on the other hand you don't "owe" her anything above ordinary gratitude to a loving mum. If you were curious about your genetic origins it wouldn't say ANYTHING about your love for your family or how well your mum and dad brought you up. I think a lot of adoptees have absorbed the message that any interest in their birth relatives is somehow disloyal/ungrateful, which is total rubbish. Your son might be geniunely curious, and that wouldn't reflect on how he feels about his granny (your mum) at all - or his gf might just be a nosy cow!
I think you should talk to your son about it, just the two of you, and be honest with him about how you feel and why you don't want him to go digging around in the past.
Good luck, it would be a real shame if it drove a wedge between you and your son. As an adoptee myself I know there is sometimes pressure to keep it all a secret and not to acknowledge it ever happened. (I am in contact with my birth family, my adoptive family have met them too, and I still wonder exactly how I will explain to my DC who all these people are and why they got three sets of grandparents...)