guesswho,
You've got a WHOLE of choices. Many years ago I read an article by John Wootters, one of my all time favorite hunting authors. The title was something like, "Eight Deer Hunting Cartridges to Start and Finish With." He listed rounds that were good for the beginning hunter, but would still be useful for an advanced hunter, as well. His recommendations: 250 Savage, 257 Roberts, 6.5X55 Swede, 260 Rem, 7X30 Waters, 7-08, 30-30, 300 Savage. Personally, I think those are all still good choices, some better than others depending on certain factors. W/ the advent of reduced recoil factory loads, there are even more good choices, including the 308 Win, 25-06, 270 Win and possibly 30-06. When faced w/ a similar decision when my 9 y/o son started deer hunting, I bought him a 308 Win. Nearly bought a 7-08, but opted for 308 because I already had all the handloading components for it. This was before the days of factory reduced recoil loads so I developed my own w/ a 125 gr Nosler BT; the load actually had less recoil than a full power 100 gr 243 load. He killed a bunch of deer w/ that load before moving up to full power loads.

Personally, I've never killed a deer w/ a 243, 22-250 or 223 because I have no desire to. I've skinned and helped track a number of deer that've been killed w/ them. Although there are alot of experienced hunters who like these calibers, I've not one of them as there's just not as much room for error as w/ larger guns. Bullet placement and performance both have to be topnotch, every time. If a big whitetail pops out at 200 yards but the only shot is a "Texas heart shot" would you want her to be shooting a 223/243 or one of the other rounds mentioned? Why use a 243 when one can use a reduced recoil load 308 that will have similar, or less, recoil, especially since your shots will be inside 200 yards?

Be sure you have a GOOD recoil pad installed. Also make sure the gun fits her WELL, especially length of pull. When shooting off the bench, encourage a "sissy bag" (thin sand bag between butt of gun and shoulder), ear plugs AND ear muffs. PAST makes a number of removable shooting pads LINK .

BTW, in the aforementioned John Wooters article, he said that instead of referring to the 243 as a women's and youth deer gun, it should be referred to an "expert's" deer rifle.