So I decided to make a replacement bulb for my dome light. I know, it has been covered here before, but I figured that I'd document as I went along and share my experiences.
36 3mm Red LEDs
6 100 Ohm Resistors
1 small piece of acrylic
Some scrap solid-core copper wire
1 Appropriately-size wood dowel
1/8" drill bit
3/32" drill bit
Pliers (needle-nose or regular or both)
Step 1: Get the overhead light outta there!
Once you've got the cover off, you should be able to see the bulb. Get it lit, and use a multimeter to check polarity and voltage. On mine, ground was nearer the switch, so I used a sharp pointed object to scribe a small "+" on the positive side.
NOTE: Be sure to check the voltage with the engine running. Most car's are 12 volt systems that are actually somewhat higher. Mine is 11.7V when the engine is off, and 13.8V with the engine at idle (and probably a bit higher at wide-open throttle).
Now you can pull out the entire dome light assembly. Again, look online for instructions specific to your vehicle. For mine, I had to pry a few clips to remove the trim, then pull the whole light assembly down and remove a plug.