The halfscreen gif is like a checkerboard pattern, alternating between a color and transparency. In your graphics program, create a new transparent image. Pick your color, and apply it to every second pixel.
I use a 120 pixel by 120 pixel image. I used my program's Pencil tool to manually do a 10 x 10 area, then copy/paste to cover the entire area. You can use a smaller image -- as little as 2x2 will do the trick -- but a smaller image will, ironically, cause the background to load more slowly, as the browser has to duplicate the tiny image many more times over.
Below is the blue halfscreen used in my ComplexSpiral page, magnified to 4x size.