To find the SSID, you can always just turn on the wireless on a laptop and see which network your home network is. If that's not available, you can connect to the router via an ethernet connection and visit the router setup page (usually 192.168.1.1 or something like that), put in your username and password (typically admin/admin or admin/(no password) or admin/password) and view the wireless settings to see the SSID.
According to the wii information page (located here), there could be several issues. Double check your SSID to be certain, also check this page to see if your router is known to be compatible.
Error Code: 51031
The only wireless network within range of your Wii console is not
configured or is not compatible with the Wii.
Various factors affect connecting a Wii to the internet with a
WiFi connection as well, including signal strength, distance from the
wireless router to the console and obstacles such as walls and ceilings that
interfere with the signal. If you're using a G router and you have your
Wii is too far away or, has obstacles in the way, it will not connect.
You should use an N-Router if there is a significant distance between the
console and the router or if there are obstacles. Here is everything you
need to know about connecting your Wii to the Internet. Make sure other wireless devices such as cordless phones or
wireless speakers are turned off, and that the Wii console is an open area
without objects or electronics blocking the signal. (For instance, if the Wii
is placed in a crowded entertainment center, try moving the Wii somewhere with more
room.) Change your router's channel setting to 1 or 11. By default, most
routers broadcast on channel 6, which overlaps with other channels and may be
weaker.
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