The real story
General Electric introduced its Snooz-Alarm in 1956. The usual explanation for the odd nine minutes is mechanical: the snooze gear had to mesh with the clock's existing gear teeth, which allowed a bit over nine minutes or a bit over ten. Nine was supposedly chosen because ten felt long enough to fall back into deep sleep. Nobody has found a design memo proving it, but digital clocks kept the nine minutes anyway.
Sleep researchers mostly agree that snoozing gives you light, broken sleep rather than real rest, and that a single later alarm is better for most people. The evidence is not all one way, though: a 2023 study found that half an hour of snoozing didn't hurt people's alertness. Either way, a regular sleep schedule matters more.