Why not make the box from plywood or even MDF? This is a common use in like say, guitar amplifier cases, PA amplifier cases. It will be a stronger box and more solid. Like amplifier cases if can have vinyl or another material glued on it to cover it, making it look very professional when done. Go to one of the music supply sites to see what I mean.

 

I still claim that I can build a very mechanically robust disco light out of a shoebox.

But the comment about fire prevention is a good one that I hadn't considered. I was planning on using fans to circulate air to cool the lightbulbs, but you guys are right. If the heat got out of control, a fire could easily be started. In light of this insight, I will now be building my disco light using a plastic box!