Without knowing anything about your problem, I suggest the following. Before you pay someone or send it anywhere, try this:

1. Unplug it from the wall. 
2. Unplug the power cord from the Wii itself. 
3. Wait 5 minutes.
4. Plug the cord back in to the Wii, and then into the wall. Do NOT user a power strip for this step. Make sure you plug it directly into the wall socket.

This not only hard-resets the Wii, but also resets the AC adapter, which is a common solution to Wii failure.

D.A.D.S. Electronics did a great job repairing my Atari 2600. However, that was in the 1980s. I'm not sure if they would service a Wii or not.

Barrett's suggestion is the best place to start.

From there, Danny G is also correct as most repair jobs are going to exceed 50% of the cost of the unit which is not cost-effective. Especially if you have to pay for shipping.

Worst case scenario, sell it on Ebay for a few bucks and use that towards a new Wii... There is a market for parts on Ebay and the device may be worth more than you would expect.

Though I would dig the Speak and Spell mod.  "Spell AB-SCESS".

Have you tried just holding the home button?

Try maintenance mode and see if you can update the system.

http://www.joystiq.com/2007/11/08/wii-maintenance-mode-its-a-secret-to-everybody/

Tried all three suggestions (thanks).  No dice.  Did some more Googling and found that it might be that because we didn't use it for a couple months that it might just be updating and we don't know it.  So, I'm gonna leave it on overnight, and then if it's dead, figure something out.

Thanks for the suggestions...keep them coming, if you have any Wii-fu to share...

*rubble rubble rubble*

I was once in a similar situation (too much $ to fix it, too broke to replace it) with a light controller that I use DJing and I brought it up to LSC, asked an electronics instructor to lead me to a promising student who might want to earn $20 bucks.  This approach not only worked they wouldn't take the $20 bucks.

Actually I am in a similar situation now with a few pieces of equipment and might try the same.

Well, it ran for 36 hours and never woke up.  So, I think a trip to PawnAmerica is in order...to get a new one, not dump the broken one.  

If somebody wants a non-functioning Wii, I know where you can get one for cheap.  :)