What do I replace a fraying drive band with?

For a spinning wheel's drive band replacement, you might try carpet warp -- a thick cotton string. It doesn't need to be waxed. Or, some "poly cord", a plasticy, slightly stretchy cord. Usually you can melt the ends together with little more than a blow dryer. The large spinning supply stores (Woodland Woolworks, The Woolery) may stock this, or you may be able to find it at a hardware store.

(posted by me this day on spinning_camp)