I think this requires a bit more information to give The Best Answer.
a) What's your budget?
b) How big (height) and strong (upper body) is he?
c) Do you live in a noise-sensitive area, or is it OK if it's a bit louder than the 10-pumper he's got?
d) Where do you live? There's always the chance there's someone local (or local enough!) to help you out in person. For example it would be very useful to have your son along "just to shoot with a buddy for a day" and see if he can cock a medium-powered springer reasonably.
Benjamin sells a Discovery kit, that includes the pump. Around $300-400, I think. Sonny Boy might need Dad's help to pump it up, depending his weight. (my boy is 12, and weighs about 87 pounds; he can't pump up my Marauder) But that's not necessarily a bad thing. He'd have 20 shots or so before he comes back to you, and they would be shots with no recoil or hold-sensitivity. If your area is noise-sensitive, you can buy a moderator for it that will quiet it down a lot.
Alternately,
my .177 Marauder is currently for sale. A nice package, and a nice step up from a Discovery. It's been for sale for a few days now with no serious bites, so if you want the gun without the scope or something to save a few bucks, we can deal. It shoots at 18 foot-pounds of energy, which is about triple what he's got now. It will handily take down a squirrel at 35+ yards. You'd just need a pump and you'd be ready to go with that.
Here's a video review I did on the gun, before I decided to sell it.
If it's out of the budget, there are plenty of good options at lower budgets too. The only question is whether he's strong enough to cock them. The aforementioned HW30s or Beeman R7 is a great gun, but not any more powerful than his 10 pumper. (thought it may be a bit more accurate) My son can barely cock it; I don't think he could cock the next step up, in a springer, which is the HW50. So to me, that points to a pre-charged pneumatic (PCP) air rifle as being the next logical step.
Or possibly a Benjamin 392, which is a higher grade pumper, .22 caliber, and only needs 8 pumps. (but harder ones) Again, upper body strength may come into play here, depending on how sturdy he is. It would be invaluable if he could try some guns out from a local member. I'm in the north suburbs of Chicago, and I'd be happy to help.