I like the suggestion to start with a film SLR (I would also suggest manual focus). This may be because I'm currently using one.
I have a history of getting engrossed in a hobby for only a short time. I've "been into" photography on and off. Recently I had a major itch to buy a DSLR. Luckily I realized that I didn't have the skill set to fully utilize a DSLR's potential.
By starting with SLR you will do a few things
1) Avoid high price tag
2) Force your daughter to learn the basics of photography. You can learn the basics of photography on a DSLR, but that full-auto mode is staring you in the face, enticing you to be a little lazy. Whereas a full manual SLR gives you no choice.
3) It gauges her interest in photography. Doesn't sound like we know how much "into" photography she is. If she is willing to suffer through the nuances of film/manual, then she might have a serious hobby on her hands. If not, then point and shoot might be the best way to go.
My decision to shoot film was based on these 3 factors - primarily #3 given my history.
While I agree that a manual SLR is a great learning instrument, I don't think it's a great way to foster an interest in a teenager. I think it may actually discourage the interest.
As a pure learning instrument, a dSLR has some advantages over a SLR -- Being able to instantly review your image, and being able to take virtually unlimited images. If you want to test different compositions and different apertures, you can do it effortlessly -- without having to constantly stop to change the film.
Plus, with the expense of film and developing film -- The $$$ can quickly become more than a SLR. Between the film and developing, you will pay about $.25-.50 per picture (and you can't delete the bad shots). I routinely take 200-400 pictures in a day of shooting. So that would be $50-$200 PER DAY. Even if you limited yourself to 2 rolls of film per shooting day, you'd still be talking over $10 per day of shooting.
Not to mention, the dSLR gives the fast gratification of uploading the pictures, emailing them, etc. As opposed to taking the film to a 3rd party for development, then needing to convert the images into a digital format.
A manual SLR can certainly be a great learning instrument. But I wouldn't "force" a teenager to learn on a manual SLR.
As to what you can get in dSLR for about $200-$250 -- It's true that you can't get a good "certified" camera for that price range. But if you look carefully on ebay (emphasis on cautious review of seller and using ebay buyer protection), you can get a respectable dSLR camera and lens in that range.
For example, I've seen the Nikon D40 go in that price range, with kit lens.
If you can get a Nikon D40 with kit lens for about $250, or a new mid-range point and shoot like the Canon S100 for about $250....
With the D40, you will still get faster auto focus, better noise performance, higher image quality.
I certainly wouldn't go with any of the very early dSLRs (from 7-10 years ago)... but the ones from about 4-6 years ago, can still out-perform current mid-range P&S's.