I've loaded reduced loads in 270, 7mm08, 308, and 30-06 to very good effect. Just yesterday afternoon a young girl killed a nice 8 point with her 30-06 and one of my reduced loads. A 150 SST traveling at 2525fps. The shot was 85 yards. One thing you have to consider is if you go with a youth rifle they can be very light. Light rifles even in very moderate calibers tend to have a good bit of recoil.Just like a 5-5.5 pound 20 gauge, they kick.
My absolute favorite rifle for a kid is an AR in 6.5 Grendel. The adjustable stock can be fitted to pretty much anyone and it's a rifle you can continue to use your whole life. I've built several for kids and they have all been very successful with them. The Grendel is well mannered, light recoil, low muzzle blast, and easy to shoot. One benefit with the AR is should they miss the first shot many times the deer will just stand there. Often just long enough for a well placed second shot to do the job. In some cases Ive had to build another one because the parent decided they liked it so much they just had to have one too. There is something about this little caliber that causes it to punch way above its weight class. If you don't want one in an AR platform they are also made in bolt guns.
I'll likely raise some hackles with this next statement but I don't care much for 223 or 243 as a deer caliber For kids. At least in the thick woods and swamps where I live as tracking can be very difficult. I've seen far too many animals lost with both. Now both of these calibers will kill if put in the right place. But I'd rather put something a little more forgiving in a kids hands.
Good response above. Like the .223 and .243 statement. 
