Wildcatter -
This is just my opinion. I think a single shot is the best way to start a new shooter as they should take things slow, thoughtful, and start developing and embedding good safety and shooting habits. I just bought my nephew a new Savage Cub, a single shot bolt rifle. It was in the $150 range. It has an aperature sight, is very accurate, has the adjustable accu-trigger, and is grooved for a scope - which we will wait for as he needs to learn how to use sights first. It is lightweight and very well proportioned for a younger shooter. I like the accu-trigger because it adds another margin of safety as well, but gives the young shooter a chance to have a decent trigger that contributes to their accuracy potential - increasing their shooting satisfaction because they can hit the target. A lot of inexpensive youth rifles have awful triggers that make it hard for beginners to develop trigger technique and be accurate. The fit and finish is very good for a rifle in this price range.
I sure wish you and your daughter the best in teaching her safe shooting. (When I was teaching my daughter how to shoot her 20 gauge single shot H & R, I took her to a public range where everyone had to throw their own clay birds. I had one of those old red plastic hand traps (hadn't practiced with it either) and when I went to sling a clay bird for her, it kind of stuck in the trap until I was way past the point in my swing where I had intended for it to let go. I skimmed the top of some lady's cap standing on the line to the left of us. My daughter did just fine with the gun handling and shooting safety. It was me and that clay bird thrower that you had to watch out for! A couple of years later my daughter got her first deer in Northern Wisconsin on a snowy morning. Ahh, good memories!) All the best to you and your daughter.