I found some soft rubber, grey in color. It comes in a roll. Was considering this since it won't hold dust and fasteners that I drop will be easy to find against the grey color. Will the magnets adhere, through the leather? If yes, I know a guy.
I took a strip magnet and sewed it to the leather.
https://www.amazon.com/Magnum-Magnetics-Adhesive-Magnetic-25-Feet/dp/B00114LU8K Cut a length and sew on one for the jaw and another so it sits on the top of the vise. I punched holes with an awl and sewed it on. Did the same with carpet strips. If you can get tanned leather, wet the leather and form it to the jaws, then attach the magnets. I just realized I need to explain this, the magnets are used to hold the leather in place on the jaws.
I used the carpet because it also absorbs any excess oil. Also small screws and parts don't bounce as bad on carpet as they do on wood. The screws and parts won't roll around either. Not sure on the rubber mat. But I have used the carpet remnants on my workbench for decades.