I don't usually ask for freebies from video game companies when things like this happen, and I don't mean to here, but I have a feeling they (Sony/MS) will do so in some form or another (even though, I really don't think they should). While I don't currently have an XB1, I do believe Microsoft would be the more likely to give away something (a game, a month of XBL, etc) due to the fact that they've seemed to double down in their attempt to get XB1's into more homes via the price drops and, unfortunately for them, this bad press (even if it's not really their fault) could erase the momentum they have been building. Even though they are despicable, I do believe that Lizard Squad might be correct in that people might attempt to return their systems, believing them to be broken. If it wasn't Christmas, I wouldn't expect any giveaways, but due to the fact there was such an influx of new owners, it might be in the best interest of both companies to offer something.
The best thing they could do is use their money and resources to hunt these assholes down that's all I want. Sure some are under age but their parents should be held accountable for their kids actions on their internet connection at the very least a monetary penalty would get the point across. Hunt em down civil lawsuit make them pay for damages because I know Sony/MS are losing money and taking hits to their brand because of this isn't free.