Your money, and doubt you could go wrong with either caliber.
FYI- The main thing here is not to set the kid up for failure!
We are fortunate to have a Marine sniper and firearms instructor in the family. Here are a couple things we did with my boys.
1. Have a good rest when practicing. Bull/sand bags are a must have. Even better is the lead sled that Bob mentioned. There is zero recoil with the sled. I like the idea of starting on the sled. They can get use to the bang and get a feel for the rifle/ trigger. Then when they are ready, transition to just the bags. Dry fire helps a ton too starting out. Every couple of shots, pretend to load the rifle, but don't. The kids learns the embarassing way about flinch, and easier for you to identify when they miss the whole target! lol
2. Form! Teach correct form! You want their cheek resting on the stock, so to move with it during recoil. If they are not on the stock, the recoil moves the shoulder first, and head last! The last thing you want to happen is the scope hitting the kid in the head/eye! Also make sure to get the longest eye relief scope you can afford! We gaged the relief for my boys, then taped paper aroung the scope(roll paper into a cone)as a distance gage. This makes sure they stay the max distance away at all times. When buying rings, you want the shortest you can find, without the scope touching the barrel.
3. Make sure the gun fits them, with a good recoil pad. (We cut the stock on our kids rifle, stocks are cheap)Then make sure that the pad is centered on the shoulder when practicing. If the gun weighs 8 lb, you want atleast half that weight pressed against the shoulder starting out. This along with the centered butt, starts the recoil up on the barrel, instead of straight back into the shoulder. Take note that they are not holding down on the forend too. You also need to make sure they are semi relaxed under recoil. Tight muscles equal bruises.
4. Ear protection is a must have starting out for the kid thats already timid of a rifle. My Nephew shot a rifle one time with my bro-inlaw, and was terrified of the bang. It took him(3 years) to age 13 to try again.
I know this sounds extreme, but in few years your kid will make you very proud.
Oh yea-IMO -The 308 wins hands down. lol