If a child is injured in a car crash as a result of the negligence of his or her family member (such as a mom or dad), can a claim be brought on his or her behalf? How is that done?

Yes, a minor child may maintain an action for negligence against his parent. See Gibson v. Gibson (1971) The claim can be brought on the child's behalf by the other parent, a guardian, conservator of the child's estate or a guardian ad litem appointed by the court. See CCP 372 -376.