This will be a very general answer to your question. If your mom has questions about traveling and custody, she may want to contact a Family Law attorney. Custody can be a very complicated area of law, and a lot may depend on the specific agreement between your mother and father.

If your mother and father have a custody order, generally the parent having custody needs to provide written notice to the other parent of the intent to move.  If the parent who doesn't have custody objects, he or she can request a court hearing and the judge will decide if you can move. A.R.S. § 25-408 states: "If by written agreement or court order both parents are entitled to custody or parenting time and both parents reside in the state, at least sixty days' advance written notice shall be provided to the other parent before a parent may do either of the following: (1) Relocate the child outside the state or (2) Relocate the child more than one hundred miles within the state."

Sometimes child custody orders are included in the language of child support orders established by the state, so if your mother is court ordered to receive chid support from your father, she should look closely at those orders and decide if she must follow the written notice provision of the statutes. This can be a complicated area of law, and so your mother may want to talk with an attorney.