In Washington, a minor 16 years of age or older can petition the court for emancipation, but it's a pretty high standard for the court to grant emancipation. Also, emancipation is often about independence of the minor, not about a minor being adopted or cared for by a nonparent.
The court will decide what is in the best interest of the child
There is no magic age when children get to decide visitation. The best interest of the child is the standard.
The law regarding a child's 'right'* to choose is a matter for each State and jurisdiction. The judge in most States, not the child, makes the decision based on the best interest of the child. Although not a standard by any means, many States have begun to give 'consideration' to a child's declaration of custodial preference when the child reaches the age of twelve or thirteen, sometimes fourteen. There are even cases when children of age 9 are allowed to testify.
The judge is normally given almost unlimited latitude in whether or not she or he listens to a child and how much weight to give to the child's wishes. In short, there is no specific "age" but the younger the child the less likely for a judge to give the stated preference much weight.
Good luck to you.
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