A lot of good posts above. Probably the most important thing to consider is whether your daughter is comfortable enough to practice with the gun until she can ensure an ethical shot on a live animal. I would take your daughter to the range and if she is still happy with the .300 after 10 rounds I would stick with it. If she shoots it but quickly gets tired of practicing, then I would go with something in the .270 range. If she doesn't like the recoil, then I would drop down to the 7mm-08 or something similar. If she absolutely hates recoil, then you'll have to go even smaller.
My daughter enjoyed coming with me on hunts, and she passed hunter safety and became proficient with a .22. Then we moved into hunting guns... She hated the recoil and informed me that she would not be a hunter. One day I brought a surprise to the range for my then 9 year old son to shoot - my grandpa's .30 carbine (essentially a scaled up .22) that he had brought home from WWII. After watching her little brother rapid fire 30 round clips, my daughter put her book down and said she wanted to try. When she finished the first clip she said, "Dad I want to go deer hunting with this". I said "okay" but I was thinking that making an ethical shot with this gun would require getting into bow range during the rifle season. We blew a lot of stalks and ambushes, but eventually caught a doe unaware at 40 yards. A double lung shot sent it stumbling down the hill and it fell after 20 yards.
My mother also hates recoil. She has harvested several muley bucks with a .223 and refuses to hunt for anything that she can't kill with that gun. Many of her hunting stories are about finishing bucks that were injured by someone with a larger caliber taking a shot that probably should have been passed.
As a fellow bow hunter, you might relate to another story. Several years ago I offered to help my cousin get his first deer. He borrowed a gun from his boss. I glanced at a nice looking gun with a big scope and figured it would do. I didn't even look at the bullets or I would have known what we were in for. We were ready to head for lunch when I glassed a buck a long way off. I made a plan and we started a long hike. Several hours later we slid into a sage brush on top of a ridge to find the buck 60 yards below us. My cousin had the gun pointed in the right direction, but said that he couldn't make out what he was seeing through the scope. I looked over and was surprised to see that the smallest magnification was 6.5x and the thing was dialed up to 25x. Open sights would have been great at 60 yards, but 6.5x had to do. He lined up again and fired a single shot. Half of the deer dropped forward and the other half dropped backward (barely an exaggeration). I had no idea what a .300 Ultra Mag would do, and I will never take anyone hunting with one again.
The caliber that I would recommend for most young or smaller bodied hunters is a .270. It is capable of taking any local big game (the world record moose was shot with a .270), but it doesn't knock a kid (or lady) over.
A 7mm-08 has about 10% less power (and recoil) than a .270. It is also a good light gun, and I would be confident with it for deer and antelope. I would also be confident with it on elk out to about 200 yards.
I wouldn't recommend the .223 or .30 carbine for most big game hunters, but if you are willing to stay within their limitations they can work well.
