I have the Fitbit Flex, and it's exactly what your mother may want. Including the Fitbit One the previous member mentioned earlier. The Flex doesn't require you to have your iPhone with you at all times, but it is required for you to view all the activity data that's being recorded by the Fitbit. You can get a rough estimate of your daily progress on the device itself, but viewing your progress is a lot better on the iPhone.
Another thing I would mention is that the band will wear out pretty easily over time. I practically wear mine at all times (except when I shower) and it's starting to show some wear. The one I currently have is black so it doesn't look too bad from a distance, but at a close distance it shows some wear. It also makes it harder to get a response from the device when you try to double tap it. When I first got it, a light double tap let me know how I was doing. Now that it's gotten some wear, I have to double tap a lot harder. Hell, sometimes it doesn't even acknowledge my taps.