I think there are only 2 possible scenario's here:
- either the young man does know his father is his adoptive father and has chosen not to contact his biological father for any number of reasons (although may do so in the future).
- he does not know about the adoption
In either case an unsolicited FB message from a 14 year old who, by the very nature of being 14 years old, is not likely to so so sensitively is hardly the way to handle this (never mind the fact the social media is entirely the wrong method of making contact in this sort of situation). In the latter situation it has the very real possibility that this would, metaphorically speaking, be like throwing a grenade into the middle of this life. You have no idea what is going on in his life right now.
If the former, the DD may well get ignored or an actively frosty reception, whilst he may be upset to be contact. The potential for this to end badly for either, or both, parties is quite significant. There is, of course, a possibility in either scenario that he is happy to be contacted- but that's a big risk to take with someone else's life, IMHO. As such, it would be a very selfish and reckless thing to do.
That said, I understand why OPs DD would want to contact her (biologically speaking) half-brother. 14 years old is probably too young to understand the potential implications of contacting this young man, but is certainly not too young to have them explained to her. She needs to be helped to understand the potential consequences of her actions and why it would be irresponsible to act on her wants/rights- although she says this is about him being angry if he finds out later, I strongly suspect the biggest driver is that she would like to get to know him.
She may not have "signed up" to the situation, but that is because it was not her place to do so. OP and her DH need to help their DD understand that, and be shown that there is a better way of dealing with this situation and not everything in right can be "fair". In this instance, his right to decide whether to contact his biological family, trumps her right to contact him because she wants to.
I would encourage DD to leave information (via you if necessary) with the registers mentioned by PP's, this would mean he would know that she would welcome contact if he decided he wanted to initiate it. I would also make one last effort to contact the young man's mum and explain that DD wants to make contact and you think she may well do so, despite the fact you have asked her not to.