The traditional, and still one of the best choices for a broom-handle bass, is what's called "binder twine." This is a (usually) sisal twine, about 1/8 inch diameter, formerly used to tie hay or straw bales together. Still used in farm country as a general purpose (read cheap) twine, but the few people who make square bales now use wire for that.

If you want reasonable "acoustic" volume, you need a string that's relatively heavy (weight, not necessarily strength) so something like a light braided clothesline rope is usually more appropriate than "skinny string." You'd like it to be elastic enough that you can stretch it a little to change pitch, but the monofilament fish line sometimes used stretches a little too much for my taste - and doesn't necessarily return when you let off the pull. With a typical broom-handle rig, you can't really change the length much - you have to rely on just changing the tension to get pitch modulation.

Too heavy a string, or trying for too much "tuning range" risks pulling the bottom out of your tub. Been there, done that, bought the tacos and ate the tea shirt.

I'd recommend looking for a light, braided, cotton or nylon twine if you're planning on recreational (and musical) use for your tub. The string isn't really all that critical as long as it gets a sound that satisfies your needs.

If you want to get into the competition for loudest/biggest/fanciest etc., then you'd maybe want to look at things like weedwhacker line or aircraft cable and such. I use lawnmower "starter cord" on my "super double (two tub) bass" but it has a finger board, and the string is pulled to a tuned tension at about 80 to 100 lb. Not something you can do very easily with a broom handle.

John