Video quality and the user's control over it are HUGE differences. On a $200 camcorder, expect little to no manual zoom (specifically optical zoom) or focus. Expect highly compressed formats like MP4 and things like that. Expect a low-quality image processor.
On a $2,000 camcorder, expect higher-quality, less-compressed, more standard formats such as HDV. Expect greater control over how your image looks, with manual focus and zoom, white balance, exposure, shutter, and so forth. Expect to be able to plug in an external microphone. Expect a bigger, better lens.
In short, the more features the camera has related to image control, the better the quality of video you'll be able to put together. The question is, which features and quality are acceptable to you? I work in video professionally, so I like as much manual control as possible. I've chosen the Canon HV30 as my home video camera for its features, quality, and its compatibility with most editing programs. I like being able to use tape because I don't have to back it up to continue recording, I just use a new tape, and tape is fairly inexpensive. For many who just want to be able to record something basic and look back at it, cameras in the "Flip Video" series work fine. It's all about your needs and how much they're worth to you. Look at reviews on sites like CNet, watch sample videos, and if you can, go try some out in stores to see what you like.
What do you anticipate your niece's needs to be? How serious a video hobbyist is she, how much does she like to experiment with technical things? This is similar to the decisions in getting a still camera - do you go with a foolproof point-and-shoot camera with few options, or do you go with an SLR camera with more controls? Similar decisions are in play here.