Why don't you just let him shoot a few different rifles of different cartridges. Who knows he might like shooting one of the larger cartridges like the 308 or 30-06 or he may really like the .243. Any of which is plenty enough for deer.

I got my first gun at 11, a 12 gague. I was shooting everything from light field loads to 3 inch buck shot loads. I killed my first deer soon after I got it still 11 years old. At 12 years old I got a 30-06. Neither of the two kicked too hard for me to shoot well.

My son is 11 years old this year. He loves to shoot my .270. The recoil does not bother him a bit. He was scared the first time he pulled the trigger then he looked at me with a big grin on his face and said "Can I shoot it qagain?".

Let him shoot a few if you can and he will tell you what he likes.