Lot depends at what range you expect him to shoot and how much you want him to practice.
My take would be 35 cal like the 350 legend or 350 buckhammer or, if your prediction for range is short, then the 357 mag.
Muzzle blast is low, recoil is low and none is flinch inducing, good feature at 9 years of age.
Also the 357 is cheap as chips to reload and what's not to like about a 92 lever gun.
Of the other 2 I'd prefer the 350 legend as there is a bunch of choices of rifles and you can load them down to 357 levels quite easily.
I have a rossi in 357 and I can load that up to better 1800 fps with 158 semi-wadcutters which can provide adequate energy levels out to 75 yards easily. The other 2 are 200 yard guns in full load format. As I said above, depends on what ranges you plan on shooting at.
I see someone suggested a 30/30 as a low recoiling option, my 94 winchester is anything but with factory loads much more perceptible than my 308 ever was, If you reload you can make the 30/30 into a low recoil outfit.
Re-reading the original post, I see you want to buy it him for practice, in that case I'd definitely say 357 mag, start him with 38 special and work up to 357 full loads if he's anything like my son he'll love it and the transition to hunting will be seamless.
Get him some really good ear protection first though.
Just my $0.02 worth.