During the Expedition Trophy winch challenge/hill climb a synthetic line broke. We had some discussions about how to execute a trail repair on synthetic lines and one evening after the competition and classes a few of us decided to have a go at attempting a repair.
Disclaimer, I'm by no means an expert but I had read about the repair technique from a link posted on the Winchline site. If you choose to repair your own winch line, you are doing so at your own risk!
The detailed repair procedure is outlined here: Trail Fix for Synthetic Winch Line. This method describes and eye splice. We used the 10' of the line that had broken to test with and managed to repair it using electrical tape, a pen, a Swiss Army knife and some patience. Since this was a learning exercise and we did not have any thread/string handy we didn't stitch the end of the line together as suggested by Samson. The stitching is very important and Sampson has very good instructions on how to do so. See above link under Technical Ressources and Splicing Instructions. In this case we were using a 12 strand rope, yes I counted them! I believe that most winch lines are of the 12 strand rope.
Another repair method described on Samson's site is the end splice should you break the line mid way. We did not attempt this repair, but it's relatively similar.
Good luck with your repair!