Get a full sized (adult) carbine. Single shot, lever action, bolt, your choice. I say carbine, not rifle, because you want something light with a short barrel; less than 20 inches. The stock will be too long. Modify it, or have a gunsmith do it for you. Wrap the stock with tape to prevent splintering and cut off an inch and a half or what ever is appropriate for your son. Use a fine tooth saw. Not a wood saw, but a hacksaw with a good sharp steel blade. Put on a new smaller buttplate to fit the new end of the stock. As your son grows, replace the buttplate with a spacer or recoil pad. When he grows some more, add another spacer, or replace the original part you cut off. Remember where you put it ten years ago. There will a fine line at the original cut, but by this time the stock will probably be dinged and scratched anyway. The line will not affect the gun's accuracy a bit. It adds character and will give him something to talk to your grandson about.