Stick with the .243 or a 7mm-08.
A .223 for an 8 year old first time nervous hunter - too many things can go wrong. In the hands of somebody who knows how to shoot the .223 is okay at best, in the hands of somebody who doesn't, the odds of wounding a deer are too great.
Caliber is important, but I'll go a step further and add some suggestions as to the procedure you use to introduce her to shooting. Then another step in caliber recommendation.
A 7-year old might be young, or small, so proceed with caution and care. Observe everything about the way she reacts and obeys instructions. (Safety first, you know.) [ol][*]Begin by having her shoot some rounds from a rifle without recoil. A .22 rimfire would be perfect. Shoot 8-10 rounds or more. Make it interesting by having her shoot at things that are easy to hit and produce some visible reaction, such as inflated balloons.[*]Next, explain the issue of recoil and then have her shoot something like a .223. This gives her an idea what to expect in the recoil department. Again, multiple rounds.
[*]Then, have her shoot a .243. She should be able to handle that without being surprised. Have her shoot 8-10 rounds. (Stop sooner if she gets tired of shooting.) [/ol]I agree with Randy's comment on the .223 -- too much can go wrong. Nor am I impressed with the almost universal recommendation of the .243 as a caliber for young people, if something else is available
. Those who recommend it usually do so on the basis of the kid's ability to manage recoil. Almost any rifle in the hands of an experienced shooter with the ability to place shots well is suitable for deer hunting, but here we're not talking about an experienced shooter. For a young child, I'd rather have them shooting a bullet with a larger cross section than .24". Instead, I'd recommend a 7mm or .30 caliber bullet with reduced velocity for a lower energy level.
So, add to the list the .30-30. A caution here: some .30-30 rifles lack a conventional safety. Avoid those. Perhaps use a bolt action if you have to. An alternative (if you're a handloader) is to load a 7mm-08 or .308 down to .30-30 energy levels and have her shoot that. The larger bullet will make a larger hole for better chances of good blood on marginal hits. She'll be well-equipped for shots up to 75 yards, and she's not likely to be capable of shots beyond that anyway.