I am skeptical of the story but, even if I weren't, I would not add a minor to the lease as a tenant if I were the landlord. If legal proceedings are required, such as an eviction proceeding, they would be needlessly complicated.

While I might be willing to notate on the lease that the minor was an occupant of the unit, I am not clear on why even that would be necessary. If the school district actually raises an objection based upon the child's surname, mom can simply present the child's birth certificate (if they had different names since birth) or proof of her subsequent name change (if she changed her surname after the child's birth), or possibly a child custody order.