Dice usually have several sides, the typical cube having six, with various other shapes in general use. Why might you want a die with only a single side? For predicaments like this:
Imagine this all too common scenario faced every day by D&D DMs around the world: your group is in a dungeon crawl. There’s a room with a pie in it, and the pie is guarded by an orc. If the characters open the door, one of these things happen:
1. The orc attacks
That’s it, there’s only the one option. So the characters do indeed open the door and the poor DM consults the chart. Unfortunately the smallest die type he has is a d4, so our DM has no choice but to roll the d4 over and over until he gets a 1 before he knows what the orc is going to do.
This is exactly the situation that 1-sided dice are designed to solve. Now the DM can grab the d1 and quickly roll just one time, see what the orc will do, and get on with the action of the game.
This is why I never became a gamer of any renown: I never could come up with neat stuff like that.
(Via this Dave Richeson tweet.)