Checking around, other people reported similar isues after receiving a Wii from Nintendo. It seems that the online works, just when it comes to send messages or receive them from your or friends it doesn't work;but every other messaging (like Mario Kart, etc.) works.
The closest to a solution seems to be.... calling Nintendo. Seems is a problem in their part and can be done through the phone:
From the N-Sider Forums:
They finally found out what was wrong with my Wii. The first time I sent it in for repair was for some problems with the graphics card. I was getting stray pixels during gameplay. Well I sent it in and upon return the pixellation had been fixed, however this new and incessant error, 110321 occured when sending messages to my Wii friends.
Well, after about six phone consultatios with Nintendo and two repair orders, they were able to finally pinpoint the problem. When my system was first sent in for repair and a new serial number was assiged to it, the repair tech. failed to assign my Wii Message Number to that new serial.
Fortunately they were able to fix the problem from thier side so I didn't have to send my Wii in again. When I checked my e-mail from work this morning I found that a flood of "Test" messages I had sent myself from my Wii came rolling in. I'm glad they were able to solve the problem as I have always placed a great deal of faith in Nintendo in regards to thier customer care.
Now there could be an alternative through homebrew or something, but at least it seems to be that the problem was that when Nintendo did the exchange of downloads; they deactivated the other console's number and forgot to re-activate it.