Ignore scs0, please.
I've had the pixelizing/graphics chip failure twice now (noticeable on the "Home" screen for VC games, parts of 'Super Paper Mario', and the title screen of LoZ), the first on my system from February that lasted six months, and then the replacement that lasted one.
The first time, I did it all via Nintendo's online repair system. I described the problem, submitted the online form, received the label, mailed it, and got it back within 14 days TOTAL. Inside was an invoice from the New York area repair shop (I got a refurbished unit) and statement that it had a one year warranty from the day I received the new unit.
It was a bit fishy at the start, as the system had a Wii console name of "SEXY". Obviously it wasn't quite wiped, and when it developed the same problem a month later, the Nintendo rep (whom I called the second time) was quite interested in the fact that the system from their New York depot still had a name. He assured that I would get a brand new console this time, and even though the repair slip said that Nintendo couldn't replicate the issue, I still got a new system (shipped to their Redmond, WA ex-HQ).
Now for his gripes:
-since you have a new system, you can't edit your Miis. My wife and son have found it's not exactly hard to remake them, though (and they actually look better since we have experience making them now).
-all your VC games will need to be re-downloaded, free of charge. I have over 30, and this was a major sticking point as I wanted a system ASAP but Nintendo told me that they would have to copy the data as opposed to the "ship a new system and send the broken one back" option.
-the first time I didn't have any important saves, but the second time all the saves were transferred with no problem.
-the Wii contact list will of course be screwed up, but it's worthless unless you exchange Miis. I can't see it being used for IM'ing, and really, how hard is it to re-enter a couple numbers? The in-game Friend Codes "shouldn't" be changed at all if your save isn't corrupted.
To be honest, Nintendo has been amazing at Customer Service for the Wii. While the repairs can't be as high as the 360 (and surprisingly that's more reliable for me with only ONE replacement) and they obviously appreciate your business.
So far my second replacement has been fine (over two months with only once or twice a week play). I've always left the WiiConnect on, and if that contributed to the issues, then it's inexcusable...since it's a feature that is hyped and discussed (and a point I made to the CS rep).