How to re-terminate frayed brake cables?
Seth's Bike Hacks channel on Youtube has a video that specifically addresses this problem:


                
            

A quote: 


  When I showed the cable fray trick in another video, some of you thought I was playing it in reverse, but it’s true! You can unfray some cables by twisting them from the base and working your way back up. This works particularly well on cables that were recently frayed, and usually by clipping a little off the end and adding a new cap, you’re back up and running without needing to swap the whole thing.