Your mother's status as an illegal alien should not prevent her from seeking justice in court. However, she should be aware of possible immigration status ramifications. She should first consult with an immigration attorney in her state of residence as well as a personal injury attorney to fully understand her rights and her obligations. She should ask regarding any time limitations (statutes of limitation) that may apply to her case.
The simple answer is yes. Your mother has a right to bring a legal personal injury action even in Arizona even if she is an undocumented immigrant. The law does not allow people to cause injury to others and get away with it based upon immigration status. Otherwise, people could be negligent, or worse, and cause repeated or significant harm to others without any civil consequences.
However, there are some clear limitations. First, no punitive damages are currently allowable to an illegal immigrant, although that does not sound like an issue in this case. Also, a wage loss claim - if there is any - is not likely going to be recoverable. If your mom does not have a right to work in the US, she is going to be precluded from making a claim for lost wages.
The question has far more complexity that this answer, and there are other drawbacks that could affect your mother's claim as a result of her immigration status, but this is the most straight-forward answer to the question.
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