Just as a child should not be allowed to choose whether or not to go to Mass, I cannot recommend allowing him to decide whether or not to go to confession. I recommend scheduling “family confession” for as often as you believe appropriate (e.g., once a month, perhaps on the first Saturday of the month) and make it a family occasion. (By the way, I don’t mean that you should shove an unwilling child into the confessional, but that he should at least be expected to go along to church with his family when the family goes to confess.) If your child asks you to take him to confession, then certainly you can arrange to go to confession at the next available time.
As for encouraging your child to confess, I recommend doing a directed examination of conscience with your children during your family prayer time and bring up sins you know he’s committed, particularly on the night before Family Confession in preparation for confession the next day. For example, when you know your children have been fighting, you might ask them to think about, “Have I fought with my brothers and sisters?” Or, if you catch your son in a lie, an examen question might be, “Have I always told the truth, especially to my parents and teachers?”
And, if your son resists going to confession by claiming he has nothing to confess, insist he go with the family to church and do another examen with him in the pew. If he still doesn’t think he has anything to confess, allow him to simply wait quietly for everyone else to finish.