Does it have to be a BB gun, or can it be a pellet gun?
If it can be a pellet gun, I’d recommend either a Weihrauch HW30S or an HW50S in .177. The 30 is a little smaller and is easiest to cock, which translates to lower velocities though still enough for small game at short distances (~20y and in or so). The 50 is a full-size rifle that’s still easy to cock, but pushes velocities up into the 700-800fps range, making it capable of small game at more extended ranges (~40y or thereabouts). Both are a ton of fun to shoot and being break-barrel/single-shot, he won’t blow through pellets too fast and will learn to make each shot count. More importantly, both are quality air rifles that should last generations if properly cared for.