He already has a .243, in my opinion the 7-08 is too close to the same power level to be a viable option (I own a .243 and 2 7-08's, so I'm not biased). I feel they are both too small for elk (kinda like giving a new shooter a .410 and sending them after waterfowl). Since you reload, I'd get him something that is a definite step up in power and simply have 2 loads (one for tons of practice, and the other for hunting). Even the hunting load doesn't need to be at the top of the power curve, but you may be surprised at how much he grows in the next year and just how much power HE will want.
I started shooting a 7mm Rem Mag when I was 13. It was a bit much for the first year, but after that, watch out.
There are so many great calibers it's tough to choose, but I'd lean towards a 300WSM, load some 308 power loads in a light bullet until he want's a little more umphh! There is great brass available, it's a short action caliber and is known for it's accuracy. It can be loaded up to levels that will take anything in north america and can be reduce loaded to very mild levels as well. 30 caliber has the greatest selection of bullets for reloading of any caliber as well.
I still shoot the same 7mm RM I was given when I was your sons age (new barrel, stock and trigger though
) and it is my favorite rifle. Get him something he can grow up with.