Can a 15 year old make it that she can live with a friend instead of at home?

That depends on the law where you live and whether the mother has both sole physical and sole legal custody of the child. If she does not, she would need permission of the court and the other parent to legally allow it and depending on the custody order, she may have to get the court's permission for such a move even if she has sole custody in both areas. She should consult her custody order or an attorney for more information.