I've created a previous tutorial here in which I designed a low-cost way to incorporate a coin cell battery into laser cut layers. For this tutorial, I was playing around with it some more, coming up with new ways to use that simple concept.
This was somewhat designed as an "intro to laser cutting class," and people could customize the lamp to their liking. Using simple graphics softwares, you can turn online images into cutouts on the lamp faces. For example, for a lamp specific to me, I had a face for the Deathly Hallows (happy birthday, Harry), a face for Naruto's hidden leaf village emblem, a face for Yuna's staff from Final Fantasy X (the good old childhood days..), a face for my golden bear pride, and finally. a face for a "Made at UMakers by Yu (since my last name is Yu.. and since it was a demo piece I thought it'd be funny if I said made by Yu but could be made by you.. get it?)
How does the light work? A coin battery is snugly fit on the side of the base, sandwiched in the layers. To turn on the light, you roll it one direction (depends on your design). To turn the light off, you roll the battery back to the original position. Simple, cost-effective, relatively robust.
And how is this done? You have two terminals of an LED, which are soldered to longer pieces of thin wire to lengthen the terminals. One wire wraps around a loop in one layer of the base while the other wire routes down to the layer underneath before wrapping around a loop in that layer below. Then the battery is fit into a pocket in a layer in between those two layers so that one side of the battery is in contact with only one of the terminals. To have the "roll forward for on, roll back for off" functionality, you have different sized loops. One loop is small enough that it'll be in contact with the battery only when you roll the battery forward.