Why does the Black Belly Whistling Tree duck sometimes stand on one foot?

Because he can. Why do you do it?

Lots of birds will stand on one foot.
Much of the time it’s to conserve heat by tucking the up foot into the belly feathers.
Our parrot will sometimes do that when she’s settling down for a nap.
Many birds have locking joints in their feet and legs so that they can settle into a position to sleep. They unlock when they pick the leg up.

Because he’s hopping mad at being given such a silly name?

Welcome to Fluther, @babydoll. I’ve never heard of this bird… but I have seen many birds do this, especially wading and shore birds. I thought it was to conserve energy while resting.

Interesting. I learned something new today.
http://www.bing.com/images/search?q=black+belly+whistling+duck&qpvt=black+belly+whistling+duck&FORM=IGRE#view=detail&id=33E9731B6197B494D5E7EE09BBD487C42BC35B0E&selectedIndex=12

A balance exercise. I sometimes do the same thing for the same reason.

To completely and baselessly speculate, and most likely being wrong, it could be a side effect of one half of the bird’s brain going to sleep, causing the connected limb to automatically retract.