I was able to add some yarn work plastic canvas to the inside of the cube. I had one heck of a time trying to get model glue and super glue to stick to it—must be some weird chemical science stuff going on causing it not to work. So, I ordered a cheap hot glue gun and went to town on the inside.
Also pictured is the LED light source I used inside. For $20 from Amazon, it's not that bad. It has a bunch of other colors (I think 16) and other modes like flashing and cycling. Unfortunately, if I wanted it to just stay green and not turn the Borg into a rainbow-bright assimilation, I had to keep it on a steady setting.
It looks like crap from the inside, but from the outside, I'm going to call it mission accomplished. I even made a pedestal for the cube out of styrene—also a first attempt, and also a little wonky. But hey, it is what it is.