No, there is not a certain age, other than when they turn 18 family court no longer controls who has "custody." The attorney for the child will meet with their children and the judge will consider their position and desires relative to their ages. The younger they are, the less their opinion matters. In general, once children hit their teen years, their desires are most often followed by the court. Children between the ages of 10-12 is more of a grey area, with some courts taking their desires seriously and others not. The particular maturity of a child is also considered; if the attorney for the child feels that their client, no matter what their age, is able to make a sound decision as to where they want to live, that attorney should advocate that position.
However, as I stated earlier, that is just that attorney's opinion. Your attorney would also advocate your position, as would the father's. The attorney for the child does not necessarily prevail in what they want for their clients.
There is no black and white answer to this. If the child is old enough to express an opinion, they'll usually be appointed an attorney. That child's preference is just one of many factors the court will take into consideration. The child's preference will be given more weight by the court the older the child is. Late teens, just about always goes the way the child desires. Single digit age, the child's preference is given MUCH less weight.
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While there is no hard and fast rule, I have found that all things being equal, a judge will start to take a child's wishes seriously once the child turns 14. Again, it is based on the maturity and intelligence of the child in question.