Apparently, the child making the threats is a well known bully with a history of similar behavior. Can she sue the school?
Edit: she is contacting the police and threatening to contact the media. I suggested she retain a lawyer. I'll circle back with updates as this unfolds.
Depending on your state, you may have state claim against school. A lot of states have adopted anti bullying statutes that give you a potential vehicle to sue school. Also, you likely have a federal claim against school. Highly specialized area of law but there attorney fee provisions so lawyers like these cases because government pays the fees.