11 year olds come in all sizes. He could weigh anywhere from 60 to 120 pounds. Makes a big difference.

I always think you can't skimp on the gun. You need to have a caliber that will do the job well. Hired pros can use some light calibers because they are precision marksmen. Most 11 year olds are at the beginner level.

243 is the smallest I'd go. If he's a bigger kid, then perhaps a 260, 270, 7-08, 7 x 57, or even a .308 if you handload. One of the newfangled recoil pads can do wonders.