I have several nieces in that size category that hunt with me, along with some young fellas, hope I can provide some advice. I have a decent selection of rifles available, and we just spent time with 2 of the smaller girls picking out the ones that they are most comfortable with.
My smallest is 11, and a little smaller than your daughter. She prefers the little break open Rossi with a youth sized stock. She fired over 50 rounds of .223 out of that today and was ready for more. If you want an inexpensive starter gun that is safe and easy to use, that's a good one. Make sure to spend the $8 for the side hammer spur, it makes cocking a lot easier for kids with small hands. I'd have preferred a .243 on that, but I found a great deal on the .223. I have another in '06 but it's significantly heavier, one of my older nieces likes that one (she shoots left handed, works great for her).
I like that the Rossis come with other barrels too, so I can swap them out and they have the same operating controls for small game hunting. You can get a 3 barrel combo in .22LR, 410 or 20ga, and 243 for under $300. In a couple years order a regular sized buttstock to replace the youth one. If she really gets into it, you can buy nicer guns later but that combo will do everything she needs and it's inexpensive and durable.
The next one up (12 or 13, about the size of your daughter) LOVES the Marlin 1894 in 44mag. It's light and handles well, and she feels like a cowgirl with it. The teen boys like that gun as well, it's built on a smaller action then the 30-30's and handles great. Her second choice is the Rossi 30-06. She's pretty tough though, a lot of kids find the recoil of that excessive.
The favorite of kids a little bigger is either a lever gun or the Remington Model 7 in 7mm-08. A little heaver than the Rossi or 1894, but still pretty light and compact, well balanced. Has a little too much recoil for some kids. One of those in a slightly lighter caliber, or with handloads at about 80% power, is a sweet gun for kids.
Almost all the kids LOVE black powder guns. Especially smaller lighter ones with traditional looks (CVA makes a couple). Traditional Hawkens and Pennsylvania rifles are way too long and heavy.We have a special black powder season here, probably not the best choice for a normal season.
Not favorites of the smaller kids: full sized bolt actions (too heavy), ARs (odd, they love shooting them on the range, but not for hunting), "long action" lever guns (the bigger kids like those, small kids find them too heavy), any stuff with long barrels (especially pump shotguns) and big optics (can't find target in the scope easily).
I'd like to see the Savage Edge youth in person, looks like a great gun for the price, but I have not had a good excuse to buy one yet. I would definitely look at that instead of the Rem 770 (I did some work on one for a friend, it's definitely built to hit a low price point, the thing just oozes cheapness. I'm not a Remington hater either, I have a few that I really like.)
Stuff that kids find important:
Gun has to be light if you are walking at all. They will be tired and dragging at the end of the day with a heavy gun, or they will always have it on the sling. Also make sure she's comfortable holding the gun up to shoot offhand, many shorter kids have trouble with that, there is just too much muzzle weight to hold up easily.
Optics: Iron sights make it way easier to find the deer in the sights. If you are going to use a scope get one with a low magnification and a wide field of view.
Sling: a must. Kids get tired and really appreciate a sling for when you are traveling between drives, walking the logging roads, etc.
Recoil: Kids vary tremendously in their response to recoil. I have one niece that is afraid of light 20ga target loads, another that can shoot 2 boxes of 30-06 in 20 minutes and come looking for more. Go a little light on the caliber, use that Remington Reduced Recoil ammo, handload light bullets over smaller powder charges, or something. You really don't want her scared of the gun or uncomfortable shooting it.
Stocks: At her height, she will have a lot of trouble with a full size stock, you really need a youth sized gun to for it to fit her well. Too big a stock makes felt recoil worse, adds weight, makes it hard to get scope eye relief right. Very frustrating to use, and the whole point is that hunting should be fun.
Accuracy: Your mileage may vary here, but I go to the range with the kids and a life size deer target (way more fun for them than bullseye targets). The max range we consider a safe shot is where they can place almost all the shots in the kill zone offhand. This year that means 30-35 yards for one of the girls and 50-60 yards for the other. A wounded animal as a first experience is a big turnoff to the kids, it can be very traumatic and you can lose a new hunter right away. Limiting range also works well when we are using lighter calibers.
Operating controls: need to work with little hands and be relatively light and smooth. Kids will almost always jerk heavy triggers and get really poor accuracy, which is frustrating for them.
We talk a lot about making ethical shots, how it is their responsibility to make sure the animal dies right away, and that nobody will be unhappy if they pass up a shot they are not sure of. Really stressing getting waiting for a good shot mostly broadside, making sure they know where to shoot, and not trying to shoot too far. We come home with no deer about half the time, but it's about the experience, not the meat.
I'd highly recommend letting your daughter handle a bunch of different guns, see what fits her, which one naturally points where she wants it too. make sure it's not too heavy. Get her a good sling and some practice time at the range.
Sorry I kind of wrote a book there, but I hope it helps. Getting kids started hunting is really important, and I'm glad to hear your daughter will be joining you in the woods.
