The best thing I have ever found for a fall is paracord. Since I make my whips from it, I have a lot of it. If you don't any type of nylon rope with kernmantle construction will work. There are many ways that people make and attach a fall, and you can look it up. I will tell you how I make and attach mine, which have never come off. A bit more time intensive than other methods though.
1. Measure a length of Paracord/kernmantle rope that is a little more than twice the length of the fall you want.
2. Remove the kern/core.
3. Using an awl or sharp non-cutting object, pierce a hole ~ 4 cm off center.
4. Using a bent paper clip, thread the end further away from the hole you just created, in through the hole and along the length of the cord until the ends are even. You should now have a 4 cm loop.
5. Cut and singe the ends together so that they are smooth.
6. If the end of your thong/lash has a hole, put the end of your fall through the hole and then through the loop.
6.5 If not, lash a thread through the loop and whip 5 times around then finish the lash as normal.
Done!
With the loop placed into the the fall and not tied, it will never come off.