what fits him best, realistically caliber/cartridge is not going to matter as hunting distances will be close for a learner so get something with recoil that he can handle and won't scare him into bad shooting habits
look at the different length of pulls on not only youth rifles but also full size rifles with shorter barrels as some offer shorter length of pull.
for an inexpensive starter rifle I just bought by daughter (12yo) a TC compass $230 after rebate
13.25 LOP, 22" barrel (except the 24" magnums), 7.25 lbs (too light and it increases recoil), 3 position safety to completely lock bolt, rotary magazine for easy/safe unloading--- and barrel is threaded so if you like you can add a muzzle device to reduce the recoil or noise--plenty of cartridge options to suit most anybody
another option I looked at american predator also-- if you are looking for a higher priced, higher quality option--browning offers 2 excellent reduced length of pull rifles --the micro composite, and micro midas rifles
a good option for a growing boy is to find a "full size" rifle and then see if you can buy a used, reduced LOP stock for short term use-- or even buy an aftermarket stock (like boyds) with a reduced LOP for starters
biggest thing is to make sure the recoil and or noise doesn't scare them into bad shooting habits
make it fun for them, offer rewards or trophies-- we shoot at oreo cookies at 100 yards, you hit it, you get to eat one--I save the 1/4 lb tannerite for her last shot-- 10-12" steel gongs, clay pigeons on the berm, small dollar store stuffed animals, a case of water bottles and a box of food coloring from walmart will run you under $10 (yes I make her pick up the busted bottles after we are done), sometimes fruit (apples, oranges, potatoes, watermelons--what ever is cheap), I make cardboard cut outs of antelope and elk for her to practice shot placement.
Keep it fun for them and they will want more -- know when to say when !!!!! (don't make the range sessions too long or boring)---hot, cold, wet, tired, sore, sunburned kids are no fun (at least mine aren't) --find a scope with longer eye relief so there is no chance of scoping
and most importantly-have fun