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.

My father just keeps an old metal tin on the stove, full of water. Not the most pretty solution, but simple.

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.

One more echo on the reduction of flow in the stovepipe. This could be awesome for humidifying the dry heat of a wood stove, but a drawback could be creosote build up. Two things in a chimney that add to creosote build up are a slow flow of exhaust gasses (smoke and tar particulate) and wide temperature differences. Evaporating water would probably do both of these at the same time unfortunately if this was put around your stove pipe... however if you could find a different heat source like the actual stove itself we could get this idea rocking.

I see two major problems with that theory. The first has already been mentioned; that being the buildup of creosote due to a lower temperatures in the stack. The second is that it would act as a huge heat sink, almost defeating the purpose of the wood stove. Many wood stoves for use in wall tents and such have a water container you can attach to the sides so you have hot water in your tent. Most of us up here never use these, as it tends to suck all the heat out of the stove, causing you to use twice as much wood and two to three times the time to get the stove up to temp. Kettles, pots of water and such on top of the stove are usually all it takes. Time tested and proven.