This may not be a problem for most people, but I have a habit of keeping drinks and food in my car so that I can use them on my work breaks. This becomes a problem in the summer when the car gets hot (140dF). Nobody likes drinking their DrPepper at 140dF. Simple plastic lunch coolers with thin foam filling and solid lids won't keep cold for long at all. After much debate over the possibility of using active refrigeration schemes, I decided that the easiest and most reliable design was simply an icebox. This would also make the cooler more portable and lend it to a variety of uses.
Step 1: Think First
As for design, we want the box to be absurdly insulative for a couple reasons. First, more insulation results in a longer melt time on a given mass of ice. That's simple enough. As long as there's solid ice, the block's temperature remains relatively constant. Second, the greater the disparity between the thermal resistance of the box walls and that of the air inside, the less of a temperature gradient exists between the ice block and the walls of the cooler. This is what allows me to avoid using shaped or distributed (bulk crushed ice) charges.
I personally needed it to fit behind the passenger seat in my car. This limited the dimensions, so I had to find a size that would work with different arrangements of beverage containers and ice containers. I chose an internal box size of 7W x 10.6L x 10H (inches). Different applications would require similar planning.
I considered a few materials for the insulation: FIP polyurethane foam inside a cardboard box, EPS (beaded polystyrene foam), and extruded polystyrene foam. Since the first two ideas suck and I already had a sheet of 1.5" pinkboard, I decided to use that, thus giving the box its absurd 3" thick walls.