I'll have to 2nd Rebel Hog's recommendations.

25 years ago, my father, when faced with exactly your decision, gave me a Remington 788 in .243. I did take 3 or 4 bucks with it (all with lung shots) using 100g CoreLokts, but none of them really dropped with much oomph. In fact, the 1st was a beautiful 10 pointer, & heran for several hundred yards before we collected him.

The .260, 7mm-08, & 6.5 Swede will all give you the flat shooting trajectory of a .243, with more energy and bullet frontal area. The recoil isn't much greater, either. Deer would be in the middle of the game-taking capability for these rounds, versus the top of the range for the .243.

Whatever you decide, getyour sonsome quality ear protection, and then get him to the range! It may take him several boxes to get comfortable with whatever you get for him.