I found this [temporary] solution at http://www.wiierror.com/wii-black-screen.php
(via email, thanks Steven)
This happened to me once, but I fixed it. Here is how:
Eject the disk
Unplug the wii at the back
Wait 1 minute
Plug it back in
Turn it on from the remote
Press reset if it does not show up
Hold the power button to shut it off
Turn it on once more.
I don't know how, but it seemed to work for me. The problem happened
when I was trying to load the Mii channel. Hope this helps !
The first time I shut down my Wii, the light went orange (as it
should). When I turned on the power with my wii remote the next day,
the wii\'s light turned green but the screen was black.
The treatment was the same as from Black Screen, No Connection:
Unplug the Wii\'s power supply unit, then replug; turn system on with
The difference is: this problem can strike in more places than just
by loading the Mii Channel. Also, I think the green light being on (yet
there being a black screen) is a common symptom as well.