A New Cheap Sax will end the joy of learning. I have seen and attempted to play and adjust a number of the less than 1000$ saxes. (I play and repair them) Their quality control tends to be pretty loose. A few of them may be OK, you might get lucky. Most of them are harder to play than a quality student model, as the pads do not seal consistantly, and next to impossible to keep adjusted even if you get a good one to start. You would be better off renting a quality student model to see if she will continue with it.

The problem with purchasing a cheap older horn is that it'll likely need a fair amount of repair work, and the problem with buying a super cheap modern horn is the quality is sometimes way out the window [though has gotten much better in the last few years]. I'd recommend either renting an instrument from a store that will offer free repairs to it's rental instruments or purchasing an entry level Taiwanese horn of known quality, like the Kessler Custom or Antigua Winds, for example. If you choose this route the internet will be a valuable resource for finding a quality Taiwanese brand.