OK, I went out to Factory Direct and looked at everything that had a string or cord. (Hair elastics, mini purses, toys...) I bought a few hair elastics/stretchy hair bands, a little mini wallet that came with a cord (I needed a mini wallet anyway), a cat toy (big stick with bell/feather dangling off it) and a $3 pair of ear buds. I also picked up a lighter at the drug store, and while I kind of wanted to tell the cashier I don't smoke, I thought telling her I was buying this to "melt stuff" might sound worse.

I tried various combinations of things but only a couple were the right size. I had thought I could trim down wider hair bands but nope, too messy and just left bare elastic threads which weren't going to be sturdy enough.

The perfect size: the wire from the earbuds! (the part from each individual earbud, before they combine to make a thicker cord.) The only problem was they were too slick to knot them, and the lighter started to char the plastic. So while this was the perfect size, and tough, I didn't have a good way (with my limited materials on hand) of joining the ends. Someone with better parts and tools could probably make this happen.

The winner: Fortunately, the stretchy material from the cat toy was the same width as the ear buds. Since the cat toy itself was just held together by knotting the thread, a very plain knot seems to be holding just fine. I did, for curiosity, light the two little ends past the knot with the lighter to see if they'd melt together. When they caught fire, though, I quickly stopped. They did leave melty stuff behind (and a smell!) that didn't exactly solidify together, but since the knot was holding fine I just cut the ends off closer to the knot.