Step 1: Materials
-Hearing Aid Batteries***
-Water (Not pictured)
*** An easy way to get watch batteries and LEDs are to find really cheap, junky toys at places like Dollar Tree or any dollar store and breaking them open to acquire the goods inside. I bought 36 "flashlight" toys(picture below) for a total of about $24. It yielded 108 watch batteries and 36 LEDs. Buying them separately would be about $13 for a 16 pack of batteries and roughly $2 per LED from Radioshack.
Step 2: Beginning
First you should check your lights by placing a couple of the batteries together and placing the correct node on the correct side to get a light of any color. Generally speaking, the longer lead, or wire jutting from the light-up section of the LED, goes with the positive (+) side of the battery, and the shorter lead goes with the negative (-) side.
Assuming your LED is working properly, you will then need to pull out a piece of tape about 1.5 inches long. About 3.8 cm. Then you will want to carefully place four of the batteries onto the tape, end to end. See the second picture.
After you have placed the batteries, you'll want to roll the tape around the batteries, securing them tightly. See third picture.
Step 3: The Hard Part
Once you have the leads secured, the light will be on now until the batteries die. Hopefully. Next you'll want to take a bit more Saran Wrap and wrap the battery stack and the light emitting part of the LED a few times. This will make a nice water-proof seal.
Make at least one more of these clusters for a better show of light. I suggest three for the best result, but two is a minimum.
Step 4: The Cup
After you've cut the cup, place the LED clusters into the smaller section. Make sure to tape down or otherwise weight the clusters so that they stay on the bottom. Add water now if you want. With the other part of the cup, tape the cut you made earlier, but tighten the cup so that the smaller base fits inside it without falling through.