Mom wants a point and shoot
Oh, and everything she will do will be in auto mode. She won't want to do anything other than zoom in & out.
Your best bet in this price range is to go over to Amazon and just get the top digital camera in your budget. Just make sure she understands that shutter lag is present at low light because of the time it takes to gather the image. At this price range, she's going to have a hard time with low light.
I'll suggest something like the Fuji S4000. It's a larger mega zoom. Since she has shaky hands, a larger camera will give her more stability. I think she's also like the 3" screen and large zooming range. Just make sure she's aware of its low light limitations.