Wellll, for a TN whitetail, a .243 will work fine. They will work on CO elk if you aim straight. Now, sorta agree w/the 280-7mag analogy...for a girl. But, there is a big dif in the amount of kick for a man. An 7-08, for a man, or woman is no big deal, for a kid, it can be a big deal.
Something that hasn't been mentioned is the rifle/caliber combo. I bought a 7-08 BLR for my daughter and it kicked way more than my bolt .270...and that was comparing her reduced loads (light bullet and the staring load in the book) to my elk loads and switching from one gun immediately to the other. Also, one of the kickingest guns I ever shot was a .243...I think a Mohawk 660 or 600. It was short, light and w/no butt pad. Set up right, there is no recoil, set up wrong, guns can be mules.
You didn't mention the age and everybody assumed what the age is (about 12, or minimum legal age to hunt). I'd buy a .243 for a kid under 16. After 16, I'd probably buy a .260. It has the same ballistics as a 6.5mm, which the Europeans use to kill just about everything up to about 1,000#. The .264 diameter bullet has a very high ballistic coeficient and at extended ranges it, in some ways, surpasses the .308 (drop, and fps).
John Wooters used to swear by a .243 on TX whitetails and he is a full grown man.
So, far, I've shot my kids .243 at elk (don't try that at home, 3 shots and I found one bullet, the same thing that happens shoulder, happens to the bullet and yes, w/the right bullet, that won't happen. After one shot, it went looking for a place to lay down (clipped a lung and hit the spine, which busted up the bullet)), yotes (it makes a softball sized hole on the back side) and antelope (fell over and started kicking DRT).
Again, if she is under 16, get the .243. It will do the job just fine, and it will always be worth what you paid for it and guess what after you see how good it works, you just may want it for yourself.