Depends on how far it goes. Kids aren’t required to carry an ID to fly domestically. And as a parent I wouldn’t carry their birth certificate everywhere. So if the name on the ticket isn’t enough it would be quite the fucking problem.
As far as the lawsuit, it depends if the tsa was being dicks, if mom was being a dick etc. If it was a quick check and mom lost her shit because she thought it was based on race, or if security blatantly said it and mom lost her shit and it turned into a bigger deal.