According to this information and this information, your error code 51331 indicates a problem with the wireless security settings. But if your brother's Wii is working and yours isn't, and they both use the same router, then I doubt the problem is with the WEP key / WPA passphrase.
When you say you have tried to "restart my router" and "reset everything and re-enter all the info", does this mean that you have reset the router itself to the defaults, or have you just rebooted it? Try wiping all the info in the router and reentering it from scratch. Be careful to keep the existing security credentials, so that it doesn't affect your brother's Wii, or anyone else using that router in the home. The suggestion in the information above that it might be a MAC address filter is also worth looking into.