Every few years, reflections begin on a new batch of trends that have made the revolution from cool to uncool to historical—and then, maybe, back to cool again. Why are those kids on Instagram embracing the Juicy tracksuits that were so gauche in 2002? What about pedal pushers? But some trends don’t get a second chance: Why, pray, did we ever spend our hard-earned money (or more likely, our parents’) on ringtones for our cell phones?

When cell phones first came out, and you finally got one, you wanted to do everything you could to draw attention to it. Custom ringtones were first popularized by a nation of early tech adopters: Japan. The Digital Minimo D319, released in 1996, allowed owners to write their own custom ringtone ditties using the keypad. It was an instant smash: A b...