While almost any cardboard box will do, I decided to build my strip light out of white foamcore. Not only is foamcore already abundant in most studios, but because it's white to begin with, it's the perfect interior for the strip softbox. I figured that a light about four feet long and six inches wide would be perfect. The foamcore on hand measured 24x48, which works out to just the right amount for two boxes.
Rather than cutting the foamcore into individual six-inch-wide strips, I flipped it over and scored the paper at six-inch intervals. This way I could fold the foamcore into shape and use gaffer's tape to support it, so I'd have the added stability of a single sheet of foamcore that remains mostly intact. The fourth strip I cut almost entirely away except for four inches at the end. This strip will serve to hide the strobe bulb inside the box and prevent a hot spot. It also helped to reinforce the box.
I could have covered every joint with gaff tape, but I decided not to, in order to save weight and keep the stress to a minimum at the joint where I would ultimately mount the box to a light.
On the far end of the box, directly opposite the position of the strobe, I cut a 6x6 square of foamcore and taped it in place to cap the end and add stability. I covered the end with silver foil, adhered with spray adhesive, before taping it into place. This foil will serve to reflect more light back into the center of the box.
If I were creating this strip light for use with a speedlight, I would simply create foamcore end caps cut to size and taped in place, then cut an opening the size of my speedlight at one end. Instead, though, I really want to use these strip lights with my Speedotron studio system, so I decided to use an adaptor for affixing a softbox to a Speedo head.
I chose six inches as the ideal dimension, which was fortuitous because the square softbox bracket ready-made for my strobes measured 6x6 as well, so it slid into place perfectly. More gaff tape affixed the bracket to the box, aiding the snug fit and ultimately delivering something that's as important as a good-looking light: a light modifier that's convenient since it attaches quickly and easily to my lights.
With the box built and the connection secured, it's time to consider the quality of the light emanating from the box. Its interior is white, which is already soft and looks quite good. If you want to go further, another option is to line the entire interior with aluminum foil. This will certainly make the output brighter and a little harder-edged than the white interior alone. A third option is to use spray adhesive to affix aluminum foil at the far end of the box—the end opposite where the light source attaches. This was the approach I thought I would favor, as this would counteract a bit of the falloff from the light being at one end of the box. Or, so I thought. I tested the strip lights both ways and decided that there was not enough difference with foil lining half of the box to justify it. The light is fairly even without any foil already, but if you'd like to up the output as much as possible, consider lining the entire interior of the box with foil.
The last step is to cover the front side of the box with diffusion material. I suppose, given that the interior of the box is bright white, you could get away without any diffusion material over the opening, but I wanted to ensure the light would be as soft as possible. (And really, if left uncovered you'd be more likely to create a harsh, specular light where the strobe bulb is directly visible.) I used a sheet of Lee half-white diffusion gel and simply cut it to size (about two inches larger than the opening in order to overlap the sides of the box) and taped it in place. Cutting notches in the corners of the diffusion material allowed it to wrap around the end of the box quite nicely.
All told, I spent next to nothing on these strip lights, as I already had the basic materials on hand. But even if you have to purchase all the ingredients you'll spend less than 50 bucks for enough materials to produce two boxes—and that's including $20 for a roll of gaff tape that's sure to be useful for a lot of other things for a long, long time. That's quite a low premium for such high value light modifiers.