Hello,
For quite some time now I have used size "A" nylon rod winding thread to make up the terminal ends on my lines and lead outs. This thread is stronger than .020 copper wire and doesn't fatigue with repeated bending. I wind the thread 3/4" down from the loop or grommet, fold the line back on itself, wind the thread back up to within 1/4" from the starting point, clip off the excess folded line, finish the wrap and tie it off with a whip finish knot. Several half hitch knots will work also. Apply a drop or two of thin CA to the wrap. Slide a 7/8" length of heat shrink tubing up and over the wrap, shrink it and it's finished. Don't forget to slip the heat shrink tubing over the line -BEFORE- you start the wrap! I use green tubing for the down line and red for up. Your choice for colors.
I use a fly tying tool called a bobbin to apply the thread. It's a device that holds the spool of thread and has a 2"-3" tube attached. The thread goes from the spool through the tube. The bobbin allows you to control the thread much better than just using your fingers and you can also control the tension by adding or relaxing the pressure on the spool with the palm of your hand as you wrap the thread. A bobbin can also be used in the same way with copper wire if you choose. Most fly fishing shops carry bobbins and thread. They also have copper wire on spools and of various sizes. Cabela's also has this stuff. Look in the fly fishing section if you might be interested in trying this method of wrapping lines.
Using a bobbin, you can speed up the process by holding the line with both hands (about 4" apart) and spinning the bobbin down or up the thread after you've started the wrap. It's the same method used for "serving" bow strings.
I've wrapped lots of lines and lead outs for myself and others using the size A thread with the method described above and have never had a failure. It's fast, strong and easy.
Joe