For $200 you won't get a DSLR (unless you buy used), so you're looking at a point-and-shoot, which is a pretty good choice for a 19-year old who may or may not "stick to" this hobby -- it'll fit in her purse and will offer a good introduction to photography. What's the difference? A point-and-shoot will give you good pictures of things that stand still (or fairly still); that is, your daughter will be able to take good pictures of her friends, landscapes, etc. -- but a point-and-shoot is no good for sports pictures. Why? a DSLR has a much faster shutter speed and can "capture" a picture faster /can recover and be prepared for the next picture faster. A DSLR also has a distinct advantage when it comes to zoom lenses; a typical point-and-shoot will have only a small zoom. Still, a point-and-shoot will make a good introduction to photography, and IF it turns out to be a passion, you can think about a DSLR later.
When it comes to point-and-shoots, I like Canon Power Shots, though I wouldn't turn my nose up at a Nikon CoolPix. This is a situation in which you get what you pay for -- you can get a point-and-shoot for less than$100, but a $200 camera will be better, and a $400 camera will be better still. The #1 feature I'd want for a beginner is image stabilization, which helps prevent blurriness.
Extras for which to budget: I like to choose cameras that aren't black or silver; this means they don't "blend in", and your daughter will always know it's hers. Be sure to include an SD card (they aren't all alike -- get a pro-speed card or fast-speed card). Even inexpensive cameras usually come with a battery and a charger; a second battery is nice, and you can buy them from ebay. And get a case to protect the camera.
Once you have a camera, encourage her to find some online lessons in composition. Pinterest is full of good tutorials. A point-and-shoot user who understands lighting, rule of thirds, and other basics will take pictures just as good as a DSLR user who just points-and-shoots. The biggest mistake I see my students making: they try to get good pictures by punching the button over and over and over, mistakenly thinking it's about catching the expression -- if they stop, think and compose a picture, they'll come up with a greater number of good pictures.
Last thought: I disagree with the above poster about phones taking pictures as well as inexpensive point-and-shoots. (Well, okay, I have an Android, not an iPhone, but I doubt the difference is great.) Phones are convenient (and "always handy" has a value), but they're not going to match the quality of a $200 point-and-shoot. For starters, they're more cumbersome to hold, so that affects quality. And the lenses just aren't the same quality, especially if you're going to enlarge images.