Constructed this way: a flat rectangular piece of wood, a little bigger than a one-dollar bill. On the two short ends, attach pieces of wood about the size and shape of playing cards, maybe a little smaller--and rounded at the top, like those funny doors in medieval dungeons. One of these sides should have a round hole about the size of a silver dollar cut into it, and a round sliding cover of about the same size nailed onto it. Here's the tricky part: you need metal screening, the kind used on your windows. Cut out a rectangle of screen that can be stapled to one edge of the rectangular base and stretched up and over the two walls. Secure it to the opposite edge of the base (and secure the edges touching the two walls) with staples or small nails.
Fill with dirt, grass, twigs, and rocks--the four basic elements of a kid's world, which also just happen to be (in any kid's opinion) the ideal microhabitat for crickets, katydids, praying mantis, wooly worms and other catepillars, beetles, centipedes, lightning bugs, and a host of other organisms Mom doesn't want to see.
Kids use many things to substitute for a true bug box--cricket cages, shoe boxes, plastic cups--but nothing beats the real thing.