I would suggest looking into the book 'Psycho Kitty?' by Pam Johnson-Bennett. There is a large section on litter box problems, and you should be able to find a copy at your local library!
Here are some suggestions from the book that may help you to pin down the cause of the litter box problems:
1. Take them to a vet to make sure the problem isn't medical (if the cat has a urinary tract infection or UTI, they can associate the pain of elimination with the box)
2. Make sure the box is clean. If you changed brands of litter, change back (or mix the two for a while). If the box has a cover, try removing the cover (my cat hates the cover)
3. Check for potential emotional causes. If the box is near their food, they won't use it. Or if they are often startled (by other pets or children), while using the box, they may choose to find a more private spot. If you have a new cat or have had a big family change lately, that could stress her out as well.
Also, you say "one of our cats". How many cats do you have, and how many litterboxes? You should really have at LEAST one box per cat.
Hopefully that sets you on the right track to finding what her problem is! Remember that the cat cannot speak to you, and this is how she communicates that she has a problem. Please don't get rid of her! It's fixable for sure!