Most digital point & shoot cameras have the capability to capture video to the capacity of the memory card.
Our Optio MX is designed more like a video camera than a still camera, has a 10x optical zoom and is quite simple to use. Likewise, my parents just bought an Optio S55 - a very small, but capable camera that you could fit almost anywhere....again, very easy to use. (add that on to the fact we also have various pentax film cameras and a family reunion starts to look like a Pentax convention!).
The best thing you can do is make a list of the features you want...like size, megapixel count, and price, then go to the store and hold several models that fit the bill, try them out and see which fits. Pretty much anything out there is capable...you just need to figure out what you want out of the camera.