Innovation.
Motion Controls which add a lot of viseral fun to some games (Sports, sandbox, etc)
IR controls which have brought point & click gameplay to consoles (put sadly not RTS... or top down WRPGs)
Arcade fun!! I very rarely (like only when I'm home sick) have hours to play a game. I usually play is short bursts which Wii is great for and the HD system generally suck at. I hate the Hollywoodization of games. I grew up on Pac-man and Galaga and that's what I want in a video game, even to this day.
Accessibility! Finally a system I can play with my girlfriends instead of in spite of them.
It's made video gaming mainstream! No longer are VG considered the domain of nerds and geeks and middle-aged men who need to grow up. Now girls, adult women and the parents of middle-aged men who need to grow up, play to!
It's small, unintrusive and a pleasant addition to the living room. I don't mind having a Wii by my TV but I hate the look of the 360 (the slim is nice). And the original PS3 was a huge space hogging beast too.
Relaxing games! And one handed gameplay! I love Endless Ocean and point and click affairs to unwind and I also play a lot while eating, having a tea/beer or doing housework. There's a lot of games on Wii I can do that with. Almost nothing on HD.
Exercise! These days about half of my Wii time is working out and keeping fit. 360 meanwhile collects dust in a corner.
Huge library of cheap games. There's a ton of great and fun games on Wii, many available for $20 or less. There's even more crap but a little research goes a long way. Anyone who says there's no good games on Wii is a snob.
Nintendo - Honestly, I'm not a Nintendo fanboy. I don't like 3d Mario, Zelda, Starfox, F-zero, hated Metroid Prime, Pikmin, Animal Crossing etc. But Nintendo makes quality titles and has made a lot of my favourite games this gen - Wii Sports/Fit, Endless Ocean, Fire Emblem, Metroid:OM, SSB Brawl, Punch-Out!, Mario Kart, Mario Strikers, NSMB, etc.
Innovative titles & Japanese titles - Unfortunately these were mostly in the first year or so, but we still get the odd gem. Titles that would probably never appear on HD because of cost and risk. Dewy's Adventure, Elebits, Sakura Wars, Let's Tap, Dokapon Kingdom, Power Pros, Ivy the Kiwi, Opoona, Zack & Wiki, etc.
I could go on... but Wii managed to do what GC/PS2 and 360 couldn't - became my primary gaming system which, until Wii had always been PC.