I would suggest you up your budget to an interchangeable lens digital SLR by either Canon or Nikon. As for the other brands such as Olympus and Pentax, I believe these use a 4/3 image sensor which is a bit smaller than the 16x24mm sensors used by Nikon and Canon. This does permit the lenses to be smaller, however it comes at the price of poorer performance in low light.
While the point and shoots have improved in respect to shutter lag, in some cases they can still pause for a half second or more after the shutter release is pushed. In addition, the single unit cameras won't permit you to change lenses to fit a specific task and don't offer the "speed" of the interchangeable lenses. Also, the unit digitals only rarely feature an image sensor larger than 8x12mm and are really poor in low light compared to an SLR. Basically, those unit cameras will be noisy and grain up badly unless you use a flash. Bottomline, the SLR's have a distinct advantage in image quality and usability with a moderate increase in bulk and cost.
The downside is the bulk, which will be greater, and the cost. A starter setup for an SLR should be doable at 700 dollars with one starter zoom. At that price point you'll get high def video capability in addition to a fine basic digital. For about 500 bucks you'll get a fine basic digital without any provision for high def video or no video option at all. BTW, I suspect the video capability at this price point will have pretty poor AF performance, it's an add on to an existing platform rather than a designed in feature. However, if you work within it's limits, it should prove to be fairly useful. Basically, video is a nice to have feature but it's not essential, so if you need to drop your budget, go stills only without fear. BTW, video will also consume memory, so if you go for a high def video option, plan on purchasing 8 to 16 gb memory chips.
As for it's exceeding your budget, tell her it's a gift for the Family, not just her. You'll get Point and Shoot ease of use, so it won't be at all difficult to get started with. Downside is that on some cameras those ease of use features can sometimes "get in the way" for really serious users.
My experience with Nikons is that they tend to overdo the ease of use features on the low end consumer cameras, on some models using full Manual Control settings requires using the manual and scrolling 3 or 4 layers deep into the control settings. What I'm telling you here is she will love it for at least a year or two. However, if she really gets serious about photography, you can look forward to her asking for a D300 or higher level camera down the road, the consumer models have a bit too much "mistake proofing" designed in for a serious photographer. I also can't comment if Canon does the same, however I expect they do. I've been shooting with Nikons since 1973 and can gladly brag that lenses purchased back in the 70's still work perfectly on my D300.
I'll also tell you to not buy into the megapixel count myth. Quite simply, a 10 to 12 mp camera will produce very high quality prints as large as 11 x 17 inches with ease. You really don't need more than 12 mp unless you plan on making very large prints. Also, at the consumer level there is not one single lens on the market that can resolve more than 12mp on a 16x24mm image sensor, so you won't ever see any improvement in sharpness unless you spend 2 or 3 grand for a Pro level lens. In addition, higher mp counts come at the price of an increase in noise or a reduction in high ISO capability.
FYI, there is a distinct correlation between image noise and the size of each pixel site in the sensor. When you try and pack more sensors in an image sensor of a specific size, you have to decrease the size of each sensor site. Make the sensor site smaller and it require more light for it to trigger, so in low light noise becomes a real issue once the sensor gets too small. With current software and technology, IMO 10 to 12 mp is the "sweet spot" of the balance between image resolution and noise at low light levels for a sensor of 16x24mm in size. This will allow good looking images at ISO settings as high as 1600 and permit use at ISO 3200 to 6400 with moderately acceptable quality. Move up to 16 to 18 mp and you'll either see an increase in noise or a reduction in real sharpness due to the use of an aggressive noise filter. To be blunt, I would avoid a 16 to 18 mp camera UNLESS it featured a larger image sensor, such as the 24x36mm "full format" sensors available in the high end semi Pro and Professional models.
Bottomline, if some salesman tells you that you just "have to have" a 18 mp camera, tell him quite clearly you are NOT interested. If he keeps pushing, tell him you would like to speak to someone familiar with the relationship between pixel site size and image noise.
Now, one word of warning. Today's consumer digital SLR's are relatively fragile, they WON'T take the knocks that something like an old Nikkormat or Pentax Spotmatic would shrug off. They also aren't truly repairable like those ancient all metal hockey pucks I referred to. However, if you treat them with reasonable care, and DO NOT drop them, they will last a good many years. It you do drop one of these new cameras and break it, the best thing to do is remove the memory card and battery and just toss it, then go out and buy a new body and learn to be more careful.