There is no magic age.
At the age of the early teen years or so a child can testify about his or her preference. But, the judge has wide discretion and latitude to decide custody using a "best interests of the child" standard.
See Brinkmeier "Child Custody: The best interest of the child standard"
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Ohio no longer has an age of election (which use to be 12 years old). Rather, the court will consider the child's wishes and concerns along with other relevant factors. It is up to you to file with the court. The court process can include an appointment of a guardian for the child, mediation, parenting investigation, in camera interview with the child, etc. These methods can all be effective in assisting the court to ascertain the best interest of the child.
Please seek a consultation with an attorney in your area. The attorney can review with you the specifics of the facts and circumstances and well as your local county procedure. This is the type of matter that can get ugly fast. Please seek legal representation to advise you properly.
There are over a dozen factors the judge is to look at in considering where a child is to live, with the child's desires being one factor. As to the age, most courts I practice in don't start to give much weight to a child's wishes until the child is at least 10 and only minimal weight to start. As the child gets older, can better articulate reasons and generally has more life experience, the judge considers the child's wishes with more and more weight.
However, the true age is 18 when the child becomes an adult and no longer is bound by the court ordered parenting plan, although child support continues longer in some cases.
If you are going to go through with it, the best method is to consult with an attorney locally who can give you an idea as to how much weight the judge in your case would give a child of such an age in making the decision.
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