I used three flourescent light fixtures, each containing two bulbs, spaced evenly throughout the Lite Brite in order to light everything up. Mount the fixtures and wire them together onto the end of an old utility plug or heavy gauge extension cord.
The bulbs are in a closed space and as such, tend to get pretty hot when you have them turned on for several hours. The ballast is probably heating up more than the bulbs themselves, but in any case, all that heat could cause problems over long term use.
To help dissipate heat, I drilled five 1.5" holes in the top panel of the case. Inside the case I mounted two 3.75 inch rotary "case" fans from McMaster
that run off of the same 110 volt AC plug. Heat buildup was a real problem until I installed the fans. Once they were in, the case ran as cool as a kitten.
Along with the lights and fans inside the case I had to install some wooden supports which hold the back of acrylic stiff and in place. Aside from the supports, the acrylic sheets are only held in place along the edge of the case, so some kind of bracing system was necessary. I used some simple 1"x1" scrap cut to length for supports and screwed them into the acrylic sheets and the backer board with wood screws. Careful pre-drilling those holes in the acrylic - it cracks easily.