It is still an unauthorized transaction. However, you now have details of who perpetrated the "fraud" so you could take that person to small claims and get your money back from them (or their guardian). Since their guardians/parents are the victims, you'd effectively be taking your customer to court.
In the end, your talking about $200. You could tell the parents that you can take the child to court to get your money back. But, it will cost you more to do that than $200 recovery, so I think your true business-case options are:
1) Refund, close their accounts, and take a $200 loss.
2) Keep the account, take a $200 loss, and get the parents to agree to their child completing X hours of community service in lieu of a civil claim. If they refuse, revert to option 1.