You can't go wrong with the .243, that said, I would recommend the 7mm-08 instead. The 7mm-08 kicks just a little more than the .243, but you will have a larger selection of bullets, if you reload. Also, the .7mm (.28cal), is a larger and heavier bullet, than the .243, which can help if the shot on the deer is not perfect or marginal. The larger bullet would cause a lot more damage to the animal than the .243. thus helping in harvesting the deer, or causing a larger wound channel for any follow up tracking. An example of this; on .243Win. the usual factory hunting load would use a 100grain bullet, on the 7mm-08, a usual factory hunting load would use a 140grain bullet.
Again both calibers are great, but if your son can handle the 7mm, that's what I would recommend.
Best of luck on whatever caliber you choose, and welcome to Handi's.