To make a very long story short, this is what I did to my Netgear router to make it finally work with my Wii. Maybe someone with a similar setup & problem will find this helpful.
Using a browser on your Mac or PC, enter "http://192.168.1.1" into the URL to get into the router. Once you get into the router's web server, use "admin" for user name and leave the password blank. (This will work unless someone has changed it from the default.)
In the left column, click "WAN Setup". Note the "MTU" value.
In the left column, click "Wireless Settings". For the Wireless Network:Mode setting, insure that "g only" is selected, and change the channel from 6 to 11. If you changed anything, be sure to click the "Apply" button at the bottom before leaving this screen.
Go back to your Wii and go to network settings. I don't have a Wii in front of me right now and my memory isn't all that great, so I can't tell you how exactly to get to it, but what you're looking for is a Wii screen that shows the "MTU" number. Poke around... you'll find it. Change the MTU to the number you noted from your router. (My MTU in the Wii was "0" and I changed it to "1492".)
Once I did these things my Wii passed the connection tests, and I downloaded the updates and browsed around for a bit on the web, so it seems to be holding a connection.
Hope this helps someone.