The options are almost limitless, if the budget is a prime consideration, go with a 20 gauge shotgun. It seems that single shot 20 gauge shotguns are pretty high these days but he'll never outgrow it. He likely wants a repeater, I know I did at that age. My first shotgun was a 15.00 (1960) single shot 12 gauge. From there to a double bbl 20 gauge and then 12 gauge pump. Still have them.
Look for youth model single shots, you can often get a new stock and bolt it on when he outgrows the youth model.