do not get her a 10.22, I do not consider them to be a beginners' 22. they can be innaccurate, unreliable, and frustrating. By the way, I have one. and it is wonderful. now that it has about 100 bucks or so, in modification, not too mention another 100 bucks or so , in extras, and goodies.
So no 10.22. Do not get a lever gun either, too much movement, to much moving at the bench, if sitting down, other problems, with takedown, cleaning, etc. What you want to go for here is simplicity, the simpler the better1!!! Accuracy is the next most important thing, as new shooters tend to like shooting better when they can actually hit small, teeneh stuff.
Start with one of two choices. marlin mod 60, semi auto.
can be had at any pawn shop for 50 bucks or so. if you do this route, make sure it looks very new and clean, as you don't know about it, neither does she. Unless you have a friend you can take with you, to check it out , to make sure it works. Older ones may have the feed throat problem, and that would ruin any fun.
the other version of this is the marlin 795, it us the same as the mod 60, but it is magazine fed, and very reliable. the mod 60 is tube fed, and will hold 14 to 17 rounds in the tube. the 795 holds 10 rounds in the mag.
Both of the above weapons are universally known to be very accurate, you cannot get more accurate, until you either step up to a cz, or get ready to spend a big bunch of money, on a more high end rifle. A marlin will even smoke a cz from time to time.
Savage also makes a mod 64, that is semi auto, that is sold in just about every gunshop or store. also very accurate and reliable. the mags are big and heavy , but allways work. the reason I don't favor the savage is, they take allen wrenches to take apart. If you were , say , on a shooting range, or in the woods, or driving around out in the backwoods, and needed to take down your rifle, you you more likely have a flathead screwdriver,(heck , even a quarter) or an Allen wrench?
next would be a bolt action, again any marlin , old or new, in the 81 series.
I have one made in 1937, and it is fabulous!!!! the new ones are mod 981's and will shoot short, long, and long rifle 22 rounds. you could buy some short, and long rounds, they are much quieter, than a typical long rifle round, because they use a smaller charge, and a lighter bullet. these will be new, about 150 dollars or less, and can be had used in gunstores, or again at pawnshops, for 100 or less. unless you know guns, allways take someone with experience , to buy from a pawn shop.
Cz again, will be a very beautiful and accurate bolt gun, for 300 bucks or less.