I personally don't care about recoil at the non-African levels, and I do care a lot about noise. At the economical end I would go Handirifle in 30-30. Skinner has peep sights for them now, though they are a tad spendy, but still not an expensive set-up.
http://www.skinnersights.com/page_16.html
The 22 inch barrel and large bore, small capacity round mean low noise. I would chuck her up in my 4 jaw and do a deep rebate to channel the blast even more away from a shooter. If you have a 24 inch barrel it is about 1/3rd as noisy at the shooter's ears as a 16 (336Y).
Speaking of recoil, small frame shooters get less of a hit from it since then tend to rock more. For me, for my 3 girls, that means teach them to shoot away from the bench and prone. Get them shooting from their hind legs, kneeling, squating, sitting. That means a rifle they can physically handle. It needs to be light, well ballanced and easy to carry in the field. For smaller people that is not the average 243 scoped bolt rifle.
I would also go 30-30 lever with Skinner sights, if I was open to a lever.
I would also consider Tasco redpoint, they don't need to be so close they will bite. They can also be mounted on an H&R. That said, I love shooting aperature sights on a rifle, so if that suits your pupose, cool. The first half of this clip shows a Scout like 700 redpoint. Second clip shows an HR with a redpoint, it's a shotgun in this case.
http://www.youtube.com/watch?v=dNG2w2TADus
http://www.youtube.com/watch?v=6p0ekR6Zk20
And if money is no object, what about a scout rifle? No scope bite. Try here at 3:30. Cooper had daughters and they all learned to shoot the scout. My only reservation is shooting the thing with such a short barrel and the noise factor. You can heal from any recoil wound, but hering loss is permanent. To loose hearing should be a personal choice, not something a minor takes on board just to be like dad or mom.
http://www.youtube.com/watch?v=RFhM0NHsYLk&NR=1
P.S. Here is a video of Clint running a Handi shotgun (1:40). It's defensive, but the point here is they can be run simply and with some speed. One can work out a single shot doctrine than will get you running and gunning, and yet you can start with a simple little safe gun, where it is obvious when it is broken, or when the hammer is back.
http://www.youtube.com/watch?v=FhgwHQCJwWw