While bolt action and lever action rifles can be fine choices, my eight year old had trouble manipulating a lever action rifle while seated at the shooting station at our range, the lever would hit the table and it was a pain. I think the bolt action might be harder for a youngster to use too.
One day I let him shoot a semi-auto pistol and he loved it. He is now ten, and shoots the pistol and Ruger 10/22 and loves it.
Safe training is key, forgive me if you have already taught him the four unbreakable rules of safe gun handling.
1. The gun is always loaded.
2. Never point the gun at anything you are not willing to destroy.
3. Keep your finger off the trigger (and out of the trigger guard) until you have your sights on the target.
4. Be sure of your target and what's behind it.
You nailed it.