Upon receiving my Wii back from Nintendo of Canada, I immediately checked the serial number. It was different. I was a little alarmed at first, until I figured that maybe they just transferred all my info onto a new Wii, rather than fixing my original Wii (it had speckles in areas with long draw distances and lots of light, mostly noticeably in outdoor areas in Zelda).
I still had to reconfigure my Wi-Fi settings and do a very large update and redownload my VC games (for free), but nearly everything else was indeed there, transferred from my original Wii; saved games, VC saved games, Miis, Wii Points, Photo Channel records, Address Book entries, Internet Channel bookmarks, Forecast Channel region settings, console nickname, well, pretty much everything I could think of was there. Except only two things (other than the serial number), one of them major.
1. My Wii console # is changed. That's okay, I only gave it to three people, so I'll simply give it out again. I can see their Miis in the Mii parade, so maybe they don't actually have to reregister me.
2. This is the big one.
I CAN'T EDIT MY MIIS! 
It says I can't edit a Mii I didn't create. This is probably because this is a different Wii, and the Wii console #s are probably in the Mii data so that they can't be edited except on the same Wii they were made from.
I have 45 Miis, four of them are family members, and me and my brother have [PRO] status in several Wii Sports. I don't want to delete those Miis' records, but what if I ever want to edit any of my Miis again? I might have to delete and recreate them!

My brother asked for the repair originally, so he's referencing the repair number as well as both the serial numbers and both the Wii console #s of both of the Wiis. Hopefully Nintendo can change all our Miis (remotely) so that they'll be editable again.
EDIT: Speaking of editing, my spelling all went in the gutter... >_<