Sounds like you're not configuring both the Wii and the router with the same info. Also, I suggest WPA not WEP as WEP sucks for protection.
Things to change on the router.
Router Password: Change this from the default. Use upper/lower case, alpha/numeric and non-standard character/s (like $#@!)
SSID: Don't use the default. Set this so it's NOT broadcast.
WPA, TKIP, 8 character key (follow password rules).
Lastly, but option is MAC Address filtering. You'll have to find the MAC Address of the devices you want to connect to the router (I don't know the screen on the Wii off the top of my head, but it's obvious once you see it). The MAC address is in the format of 6 sets of hexadecimal numbers. Example: 00:1a:3c:af:d5:de
Once you have the router configured the way you want, you'll need to MANUALLY configure the Wii with matching info (SSID, WPA, TKIP, Key). Then it'll connect on it's own.
The only reason you should use WEP is if you have an old WiFi device which only allows for WEP (like the old DS's).
The most likely problem you're having is that you enabled encryption (WEP). When you do this, you need to manually configure the client (Wii, laptop, DS) with the key used by the router. My guess is that you didn't do that.