Winchester makes a few youth models in 12 gauge that hang under/around $1000. If you don't mind the 20 gauges and don't want to spend a lot of money, Tradeex has Armsans which you can find in more brand name of mossberg and weatherby and there are Tristars as well that come in a variety of shapes and colours. There are reduced fit berettas around but they are rare. Browning does make youth fit guns as well. If you want to break the bank, there is the Syren line by I think Fabarm.
I am currently building a wingmaster with a short barrel and adjustable stock for my fiance to get her into trap and hopefully eventually skeet and sporting.
As others have said, it has to get her excited and she has to like it. I tried several smaller and lighter full size 12 and 20 gauges and she said they were too front heavy, even when the LOP was reduced to 13.5" (I added an adjustable butt plate as I prefer a shorter LOP myself) on my beretta 391. Just take her to try some, you may be surprised what she likes. I had to talk her out of a stoegar competition at a local shop, it was not in the best shape.