Step 1: Recycle Instead of Dump
I decided to recirculate the water back into the humidifier instead of dumping it.
Start with two 5-gallon buckets. Tape the buckets together, bottom-to-bottom, and drill a pilot hole through the bottoms of both buckets. You might also want to remove the handle on the bottom bucket because it tends to get in the way.
Now pull the buckets apart and use the pilot hole to drill a 1" hole in the top bucket and a 2" hole in the bottom bucket. Also drill another 1" hole in the side of the bottom bucket about 5" from the bottom. Check out pictures in the next steps first and you will see why these holes go where they do.
Step 2: Parts You Need
You probably want the type of toilet fill valve that has the float riding on the valve itself, rather than the old-fashioned type with the stalk and balloon float. The old kind is what I have and you'll see what I had to do to make it fit.
The hard part for me was figuring out all the connections and going from pipe to hose to smaller hose. It took a few trips to the hardware store.
Step 3: Putting It All Together
Connect the pump to the humidifier (I pulled the hose off the humidifier's water supply solenoid and connected it to a barbed fitting on the pump hose).
Wire the pump to the heater. You probably want to use the heater's "EAC" connections, which provide 110VAC while the heater fan is running (presumably for an Electronic Air Cleaner?). It's best if you can wire through a switch so that you can turn off the pump in summer or when you're not running the humidifier.
Finally, drop a bleach toilet cleaning tablet into the bucket and put a cover on the bucket. This water is going to be going into your air supply so we don't want to be giving anybody Legionnaire's Disease.
Final note: if I hadn't already paid for and installed the humidifier, I would have searched for a water-saving humidifier instead of going to all this trouble. This only makes sense if you're trying to improve an old humidifier but better ones are available now.