I've thought on this subject a bit.
First off, it needn't be just one, obviously.. Maybe two or three or four, depending on how many kids there are and budget.
If it is one, I'd make it a centerfire turnbolt or a manual repeater .22lr rimfire (levergun, pump, or turnbolt).
In a centerfire, Cooper, Sako, Kimber, or full custom of some kind ideally (or at least Browning A-bolt, Winchester 70, or Wby Mark V) with an interesting wood stock - bang the stock up while you're alive so that the kid won't be afraid to use it. Chambering - think .243, .257 Robts, .25-06, .260 rem, 6.5x55, 7x57mm, .270, .280, 7mm WSM, .30-30, or .30-06 - something in that general 6mm to .30 cal range in more standard or classic chamberings. Oh, turnbolt or levergun. I don't think that a .300 maggie is a good choice (ever, for anything, but especially this purpose).
In a rimfire, something like a Browning BL-22, Marlin 39, Rossi 62 pump, Henry pump (espec. the yellow boy ones), or CZ 452 / 455 turnbolt are nice yet fairly inexpensive (all sub-$550) legacies.
Lots of other pretty good choices too. Browning BLR centerfires, Anschutz rimfire.
Go to the Mauser website and look at the new Mausers.
You can get the economy model there for around 1200-1500 bucks, and it will far outshine the any of top models to be found in the US. The upper-end Mausers are almost too beautiful to describe.
Thank you for that info, and here is the link:
http://www.mauser.com/Rifles.61.0.html?&L=1
However, on the M12 and the M03, the two new designs of turnbolts, they went with detachable mags on two otherwise-impressive designs -sad. Detachable mags are a detestable abomination, particularly in those very expensive Meister Klasse M03 where said detachable is impressively engraved. They took a great idea and ruined it with detachables, just as several American makers have done (X-bolt for one).
Thankfully, the old standard 98 model is still available and doubtless well-made by the original Mauser Jagwaffen in Deutschland.