UPDATED VIDEO AT END OF GUIDE! Just thought I'd share my mod. I hope it may inspire some of you to try something, too. 
Before you can begin, you'll need to remove "the cubby" from its home in your dash. Since writing up these steps would be redundant to ebumdude's copies of Crutchfield's Mastersheet, I decided instead to make a video showing you how to remove the cubby as well as addressing some issues you may encounter during the removal process.
NOTE: In case the content here ever "breaks," this can be found at http://mc3w.com/!/lqc/, too.
Now that you have the cubby free from your car, you can get a better idea of how you will run your wires, where to mount the LED's, etc. I have a bunch of spare LED's on hand, so the following includes the steps I took to make my own LED strip for this application, but you can purchase one from a retailer and achieve similar results.
Looking inside, you'll notice that the top of the cubby is curved as well as having a beveled area running along the middle from front to back. If you're using a flat LED strip, mounting them in this direction would probably work best.
The beveled center worried me a bit, but the curved surface was looking to be a bigger hurdle until I looked over at my trash can and saw the packaging material for my just-installed LOC. Perfect!
Cut and trimmed to fit as well marked locations for LEDs mounting holes
Using a screwdriver roughly the same size (diameter) as the LED's, in this case 3mm, and a butane torch, make the holes and then trime off any remaining plastic so that you have a nice, clean mounting location.
Now to wire up the LED' s to work with the car's 13.4v. To keep them from burning out, we'll need to include resistors to the circuit, and to figure out which resistors you'll need, head over to http://led.linear1.org/led.wiz. Just enter the appropriate values into the calculator, and it will generate a wiring diagram (or more) that shows what resistor(s) to use, and how to properly connect them to the LED's. Once you've got it wired up, be sure to test the circuit before doing any (semi-)permanent mounting. If you have a breadboard, this would be a good time to use it, since you can lay out your LED array without soldering. If everything checks out, the LED array is ready for mounting to the base...but before all that, it's off to paint.
After painting, I simply used a clear silicone adhesive to mount my LED array to the base. I chose this because it will keep them on there quite well, I won't have to worry about short circuits, and...well...it's just what I happened to have.
You really can just any adhesive you want, but if you do use something like silicone, remember that you'll need to give it some time to cure. I chose to do a test mounting inside the cubby while I was waiting for the silicone to cure. This also assured me that I didn't mess anything up during the application of the silicone.
...and a few hours later, the silicone is now stable enough (24hrs is recommended for a full cure) for me to install the cubby back in the car. Below is a photo of the final product. The video below it shows the LED's in action.
Before you can begin, you'll need to remove "the cubby" from its home in your dash. Since writing up these steps would be redundant to ebumdude's copies of Crutchfield's Mastersheet, I decided instead to make a video showing you how to remove the cubby as well as addressing some issues you may encounter during the removal process.
NOTE: In case the content here ever "breaks," this can be found at http://mc3w.com/!/lqc/, too.
Now that you have the cubby free from your car, you can get a better idea of how you will run your wires, where to mount the LED's, etc. I have a bunch of spare LED's on hand, so the following includes the steps I took to make my own LED strip for this application, but you can purchase one from a retailer and achieve similar results.
Looking inside, you'll notice that the top of the cubby is curved as well as having a beveled area running along the middle from front to back. If you're using a flat LED strip, mounting them in this direction would probably work best.

The beveled center worried me a bit, but the curved surface was looking to be a bigger hurdle until I looked over at my trash can and saw the packaging material for my just-installed LOC. Perfect!

Cut and trimmed to fit as well marked locations for LEDs mounting holes

Using a screwdriver roughly the same size (diameter) as the LED's, in this case 3mm, and a butane torch, make the holes and then trime off any remaining plastic so that you have a nice, clean mounting location.

Now to wire up the LED' s to work with the car's 13.4v. To keep them from burning out, we'll need to include resistors to the circuit, and to figure out which resistors you'll need, head over to http://led.linear1.org/led.wiz. Just enter the appropriate values into the calculator, and it will generate a wiring diagram (or more) that shows what resistor(s) to use, and how to properly connect them to the LED's. Once you've got it wired up, be sure to test the circuit before doing any (semi-)permanent mounting. If you have a breadboard, this would be a good time to use it, since you can lay out your LED array without soldering. If everything checks out, the LED array is ready for mounting to the base...but before all that, it's off to paint.


After painting, I simply used a clear silicone adhesive to mount my LED array to the base. I chose this because it will keep them on there quite well, I won't have to worry about short circuits, and...well...it's just what I happened to have.

You really can just any adhesive you want, but if you do use something like silicone, remember that you'll need to give it some time to cure. I chose to do a test mounting inside the cubby while I was waiting for the silicone to cure. This also assured me that I didn't mess anything up during the application of the silicone.
...and a few hours later, the silicone is now stable enough (24hrs is recommended for a full cure) for me to install the cubby back in the car. Below is a photo of the final product. The video below it shows the LED's in action.
