If you are using a single-walled chimney, you could saw some cuts in the bottom of a bucket, bend the metal flaps down and slide it over the chimney. Use rivets through the flaps to fix it to the chimney. Use sealant to make sure the hole in the bottom of the bucket and the rivets don't leak, and fill the bucket with some water.

Another option might be to wrap some fireproof cloth (e.g. fiberglass) around the chimney and let drip water on that. This will let you control the evaporation rate.

But the simplest option would probably be to hang up a bucket with a small spout next to the chimney, so that water from the spout drips onto the hot chimney and evaporates.

Note that unless you use demineralized water you will get scale deposits on the hot surface.

Also note that whatever you do will decrease the draft of the chimney somewhat, since you're diverting heat to evaporate water instead of expelling flue gases.

And if you have a double-walled chimney, all of these solutions will have significantly lower performance.

Regarding Joy's comment: That's what my Mother used to do when I was growing up in Brooklyn, NY. She'd put a pan on the radiator and fill it with water so there was some type of humidity in the air.

I have friends who heat with wood and they just keep an old cast iron kettle on the stove filled with water.