I'll preface this with that I'm a Hunter's Ed instructor and I get questions like this a lot.
Fist, find out if your state has a minimum caliber requirement for deer. Some states forbid using .223 and smaller.
Second, a 22-250 on a deer at 300 yards... I would not do this at 300 yards, or at any range. My personal preference, and I'm sure someone will be offended. Personally, I think the .223/5.56mm/22-250 and related calibers are too small to ethically and humanely kill a deer, even at close range.
I'd recommend looking at a larger but mild recoil caliber such as the 7-08. He'll be able to take just about anything on North America with that.