My bet it is one of two things.

One, he just likes archery better (there are folks out there like that).

Two, almost eleven? I'm afraid you're a bit behind the curve. The gun is probably going to have to go "bang" before it is going to gain much interest; the bow and arrow has more power and perhaps more range (sshhhhhhhthackk!) than the BB gun (snap-tap-ping). If he is interested, then turn the kid loose with some real firepower!

Good place to start is 22LR's into a dirt bank where he can see the impact of every shot -- Ruger 10/22 with a full clip (towards the end of the session) would be tantalizing, shotgun with trap loads (hand thrown cans [with dirt in them] are easier to hit than clays), .223 into jugs of water off of a rest makes things interesting -- give the kid some exposure to something with some adrenaline to it. In ones youth (yours, theirs, mine) there is a definite attraction to firepower -- not magnums, just something with some sex appeal to it.

Nowadays one of my favorite guns is a 5mm air rifle, BUT ONLY after that I had already been down the road from 22LR to 416 Rigby and .410 to 10 gauge.

My $.02.

Good Luck and Good Hunting,
EKM