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I have two sons who started at 10 hunting whitetails. They have both killed 5 or 6 so far. They have used .243's since day one. I originally bought a H&R single shot in the youth model. Great shooting little gun that you can scope and have ready for under 300. They not shoot the ruger M77 compact in stainless synthetic.
I am an overkill gun hunter myself, using 7mm Mag, 300 WSM and occasionally the 300 Rem Ultra Mag. The 243s have killed as well as any caliber I use. Hardly ever chase one they shoot.
I would put a little thought into bullet selection though. We have tried several. The least impressive being the ballistic tip. Gun is to fast and bullet usually explodes, doing plenty of damage, but not exiting. Which does not leave a good blood trail if you do have to track. We are now using a 100gr soft point by federal. it does a great job.
I think the key will be getting him comfortable with what he shoots. Teach him to take his time and make a good shot. Most of all knowing what is around him and behind the deer. I remind my sons everytime we go out. Safety is the most important thing, then have fun!!