This is easier than you think. Most of these kids have no clue what they're actually doing and link their personal identities with the name they use.

I run (as a hobby) some game servers and have handled several DDoS incidents. Out of all of them, I think there's only been one that I couldn't find real-life details for. When this occurs and the perpetrator is a minor, and I can find details for them, I give the parents a friendly call and let them know. Usually this can be easily resolved in a peaceful manner and without any mention, threats or use of legal action.