That would largely depend on a multitude of factors: If you represent yourself and you have her consent, it would only cost you, likely, a filing fee. However, if you have her consent, she can just let the child live with you for an extended visit and you need to have court intervention.
If you want to fight her for custody, and use a lawyer, it would likely cost anywere from $10k upward and it would depend on the number of motions made, the cost of expert testimony, such as psychological, etc.
Keep in mind, given the fact that she can get state assistance, despite being 'poor' - I find it doubtful that you would win a case on that basis alone, if she is an otherwise fit mother.
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