Just stumbled across this site, and here are other (great minds or fools, you chose) discussing a bike trailer for camping/sleeping. I first considered this topic when I was about 8 years old. Then later when I met a guy touring the USA with a total of 200 pounds of bike, trailer and luggage. I asked why so much stuff and he simply said it was a yearlong journey so why not bring a bunch of stuff. He figured that he put the same effort going 60 miles with the full load as an unladen rider would do for 100, but he had an 8'x10' tent, ice chest, and cot.
So on to the design details.
No solid floor. Just look at most old style camping cots. They're a "trampoline" of a single piece of fabric held by an exterior framework. Do the same with coated nylon cordura and you're set. You would need to be able to have enough tension but otherwise it's light and weatherproof. Maybe just buy a cot to get started, then waterproof it with a layer of light coated ripstop underneath.
For walls and a roof I also have a suggestion- I've seen several applications of basically "plastic cardboard". Built just like the corrugated cardboard that boxes are made of, but plastic instead of paper. Stiff yet flexible in one direction, translucent for light, and provides decent insulation- probably about R-2. Aluminum tubing would be needed for a framework and then rivet and glue the panels on.
If the whole goal is just a quick-setting, bugproof and rainproof place to sleep off the ground, how about the Kamp-Rite Tent Cot from
http://www.pahaque.com ? I have two of their products (TePee and Screen Room) and their designs and quality are the best I've seen in the tent industry.
Slap the Kamp-Rite down on top of a Burley cargo trailer and you're set. (Folded I mean)