I use the Keeper (which is similar to a Diva), and like it. I don't know that it would be the best choice for a young girl, though. I used mine for about a year before I got married (and, yes, I was a virgin on my wedding night) and had no problems, but I was 26 when I got married, so that might make a difference.
I would recommend that your daughter stop using tampons, if she doesn't change them regularly -- no point in risking TSS! The Keeper (and perhaps the Diva cup and the Moon cup, and whatever other menstrual cups there are) offers a 90-day money-back guarantee if you don't like the cup, so you might get one and see what your daughter thinks of it for a few cycles. I'm not sure why pads wouldn't qualify for "simplest, healthiest" -- I think they're healthier than tampons, although I have nothing necessarily against tampons -- it's just that the blood leaves your body instead of staying inside, which sounds better to me. And it's a lot simpler to change a pad than to change a tampon or a menstrual cup.
The cup fills up with blood, which you have to dump into the toilet. If it overflows, it can get messy in there (and in your underwear) -- if your daughter hasn't been changing her tampons regularly, she may not change the cup regularly, and end up with bloody panties. If I think I might not get a chance to empty the cup and/or it's a heavy flow, I wear a pantyliner as well. Inserting the cup may get you bloody fingers (which can be wiped off with toilet paper in a bathroom stall, so it's not obvious when you leave and wash your hands) when you insert the cup the first time every cycle, or if you overflow.
Once you get the hang of it, it's no big deal, but some girls may be grossed out by it or something.