Re: 4H small bore .22
Look at the local gun store used rimfire rack, and talk to the owner about your specific needs. Most good shop owners will understand those needs and take a special interest in helping a kid get started. They may not have exactly what you need on the rack at the time, but they can call around and find something you'll do OK with. A reputable dealer will not screw a kid.
A good starter rifle is a bolt gun, preferably a single shot, and has iron sights. The make is not especially important. Better to get an adult sized stock than to stick yourself with a kid stock he'll outgrow all too soon. As his first rifle, he'll want to hang onto it when he's old and gray, and it helps if he can still shoot it. There's something classic about watching a young 'un wrap his arm around the stock under his armpit and shooting the Dickens out of a paper target.
Try to ensure the crown is in good condition; no nicks, deep pits, or major rust spots where the muzzle and rifling come together. Thankfully, a crown job is not particularly expensive and a good gunsmith can handle the task very easily. Even if the crown job cost rivals the used rifle's purchase price, it's a good bet it's worth it; as long as the rest of the rifle is sound.
Greg