Here's my suggestion: Marlin 336Y Spikehorn .30-30 or 1894 in .44mag. I'm not an overly tall guy, 5'11", 175#, my lil'lady brought me a .30-30 home the last week overall length is 34", bbl is 16", GREAT accuracy, it's a little short on pull, but it's a youth model, what should I expect, but since it's so small overall, you don't notice the short pull (better than long anyway!!!).

The 1894's are great, I don't know that there is a shorter youth version, but shortening a stock is easy, and they're really short anyway 37" so they balance very well. The .44mag isn't necessarily a beginners cartridge in most peoples eyes, however, as far as field performance goes, it's every bit as potent as the .30-30 (and occasionally even surpasses it on paper too!)

Although my hatred for winchester 94's makes me wish I couldn't say this, but they've got a 94 Trapper in .30-30 that would work well (DO NOT GET A WINCHESTER PISTOL CALIBER 94!!!!!), I owned one for a while in .30-30 and loved it, but my hatred for winchester leverguns forced it out of the house.

Leveractions are great beginners guns, and they're more fun to shoot, not to mention how mild they are on the shoulder in .30-30 and .44mag, they're light, compact, and every bit as deadly as their bolt action brothers

Otherwise a .243 NEF single shot would be good, teaches discipline and accuracy, and they're cheap enough that if he drops it in the mud, who cares?