After a few days with the Wii, we discovered that the Internet had quit working for the Wii. We tried all of the stuff at the support site–changing the router to broadcast at channel 11, entering a static IP address, all of that. None of it helped.
But we did get it working. It started working again when we removed the old Gamecube memory cards from the back of the console. Once they were out, the internet started working fine.
Hope that helps some Googler out there!