In the meantime...I'm trying to find a digital camera for her that doesn't look like you need to connect the dots in the pictures.

Any suggestions for something in the $50 range? Or does this simply not exist?
I teach photography, and I'm going to go with no, what you're describing does not exist. A digital camera in the $50 range is going to be very low quality and will give you low-grade, grainy pictures. A camera in that price range will not give good color saturation, and it will have a low-quality lens (so it'll be easy to get blurry pictures unless she holds very, very still).
However, for a child -- especially a child who's hard on things -- it might not be a bad choice. If she just wants to pop out images of her friends and won't particularly notice quality (and lots of people don't), then go ahead and get it for her. You might make a deal with her that if she proves herself with this one, takes care of it well for X amount of time, she can "move up" to a better one later.
Advice for shopping in this price range:
- Don't be afraid to look at refurbished cameras. They're budget-stretchers, and they're likely to be just as good as a $50 camera.
- Go for a model with AA batteries rather than a recharger. This'll cost less, and I think the ultra-small cameras with built-in rechargeable batteries aren't "quite there yet" as far as technology goes.
- Definitely look for an older model. Yeah, it won't have all the bells and whistles, but it'll get you more camera for the money. Other people have suggested Kodak Easy-Share. It's been around a while, so older models are definitely available. I have several myself (bought over a number of years), and all have been decent cameras.
- Pretty colors tend to cost money.
And consider this: For $50, you could have 10 disposable cameras. How long would that last her? Aside from the environmental impact, you wouldn't have to worry about her destroying the cameras, and the photograph quality may be better than a $50 camera.
And a deal that may no longer be available: Back when I first started teaching photography -- that's been 9 years now -- both Eckerds and CVS had a "kids camera" deal. You bought a camera for either $10 or $20 (more expensive version had a flash, making it good for both indoor and outdoor pictures). It was a film camera. When you "took up" the pictures, you returned it to their store and had the film developed. As long as you had the film developed with them, they'd give you FREE film and batteries for as many times as you brought the camera in. You couldn't change the film yourself because opening the camera required a special tool.
Of course, that's also back when the film developing people used to pop the AA batteries out of the disposable cameras and sell them for .10 each. Those deals may be gone-gone-gone by now.