Summary

Light up your TARDIS off of a single AA or AAA battery! This design runs off of a joule thief to light up your LEDs off of a single 1.5v battery. It can even run off of "dead" batteries that don't have enough juice to power other devices.

This does require some soldering in tight spaces, so it may not be for the faint of heart.

Instructions

You'll need to know the basics of how to make a joule thief. A great instructible can be found here. The center circle is sized for some toroids I ordered from Amazon. Other than the materials needed for the joule thief, you may also want to pick up some battery springs and plates. The ones I used can be found here on Amazon. And, finally, you will need to find a SPDT (single pole, double throw) switch and jumper wire. You may also want to have a hot glue gun and glue on-hand to secure things into place and help prevent possible short circuits.

Once you're comfortable with how the joule thief goes together, you're ready to proceed.

Solder some 1" jumper wires (x4) to the positive (plate) and negative (springs) battery spings/plates. Try to keep the soldering as light as possible to make it easier to insert into the channel for the springs/plates.
Solder the other end of the 1" jumper wire on the battery springs to the outer pins on the SPDT switch.
Solder another 1" jumper wire to the center pin on the SPDT switch.
Solder the jumper wire from the center pin on the SPDT switch to one of the single wires from the joule thief.
Solder the jumper wires from the battery plates (positive side) to the double twisted wire on the joule thief.
Insert the battery springs into the 1mm channel on the negative (-) side on both battery sides.
Insert the battery plates into the 1mm channel on the positive (+) side on both battery sides.
Push the SPDT switch into the hole in the base wall on the negative side.
Prepare ~6" of jumper wire to be used to run to the LEDs in the TARDIS.
Finish wiring/soldering the transistor, resistor, and 6" jumper wire on the joule thief. I highly recommend twisting wires together before soldering and constantly checking to make sure everything works. The less you have to undo, the better!
(Optional) Hot glue the contacts on all the connections and add some extra glue to help hold things down. You may also want to add some glue in the hole that the SPDT switch is placed in. The tension from the jumper wires may be enough to hold it in place really well, but a little extra hot glue never hurts. Once the hot glue has cooled off and dried, cut away any glue on or above the 1mm shelf that runs around the center of the base.
Place the cover over the center of the base (to hide all the circuitry), but make sure your 6" jumper wire is aligned in one of the notches to run into the main cavity of the TARDIS. You may have to sand the edges of the cover so that it will press-fit down onto the 1mm shelf and stay secure without bowing. It's design to be friction-fitted, and I would advise against using any glue to hold it in place in case you need to fix/solder/glue stuff under the hood.
Position the LED or LEDs you wish to use inside the TARDIS cavity. A 5mm LED seems to easily slide into the lamp shade at the top and helps hold any other LEDs up and in place. I wouldn't recommend any more than 3 LEDs just to keep things from getting too dim (you are running on a "dead" battery after all!).

The circuit is designed so that it only runs on a single AA or AAA at a time. I'm not an electrical engineer or anything, but something about mixed voltages and remaining capacitance seems like it might not be optimal. So just leave only one battery in there at a time. Assuming only one battery, when you move the switch to the other side to turn the circuit off. Also make sure you don't glue the TARDIS body to the base and make sure there's just enough slack in the jumper wire in the cavity so that you can open the TARDIS to swap out batteries (see picture).

TARDIS Battery Base for Joule Thief