What follows is just my experience. Your mattress and situation will be different so think of this as a sketch and use it as a jumping off point. If you have a better idea that's great. My partner only decided to post this after I was finished so it's not as complete as it might have been.
The overall plan is to cut out a middle section of both the mattress and the box springs leaving two ends for each of the same size. Then closing up the ends and sewing covers for them out of canvas and upholstery fabric. The basic idea is pretty simple but it is a bit of work and requires both a Dremel tool (or something to cut the metal springs) and a sewing machine.
I used a Sealy Posturepedic Full mattress but yours will probably differ in brand, size and construction. I'm only describing what I found and did. Be flexible as you will probably have to make adjustments depending on what you discover after you open yours. I recommend a little exploratory surgery and poking around inside before you make any major cuts. But the basic approach should work for most mattresses.
Purchasing the fabric is the most expensive part of this project. I used remnants I already had so this project cost me nothing but your situation will probably be different. Note that upholstery fabric can run $30-$40/yard so keep your eye out for sales. I also used canvas for the sides which can cut your costs greatly. You can also do one side using less expensive fabric and the other in a nice upholstery fabric and flip it over for company.
Note: Don't forget to use goggles, ear protection and gloves.
Dremel tool with a fiberglass reinforced cutting wheel
Sewing needle and upholstery thread
A lot of straight pins
hand clamps (the ones that look like industrial clothes pins)
bee's wax (to wax hand sewing thread)