I literally came up with this idea because I’m tired (I had trouble falling asleep and then woke up early, it was a whole thing).
Sleep comes from the Old English slaepan (for the verb) and slaep (for the noun), both meaning sleep, and before that the Proto Germanic slepanan. So I guess each iteration just gets rid of an “an”. It can be traced all the way back to the Proto Indo European sleg-, which means to be slack or languid and has a lot of weird descendants that are either related to slacking (like lax, release, and languid) or just make no sense (lease, relish, and catalectic). I could probably do a whole post on these weird words (and I’m sure I will in the future).
Doze showed up in the mid seventeenth century (possibly earlier in some dialects) and is thought to be Scandinavian in origin, although it doesn’t get more specific than that. It also might be related to the Old English word dysig, which just happens to be the origin word for dizzy. Fun fact, dysig actually meant stupid.
Slumber showed up in the mid fourteenth century, although it also existed a century earlier as slumeren, related to slumen, to doze. I guess they needed to use something before doze was invented. It’s thought to be from the Old English sluma, light sleep. No idea where the b comes from. It just seems to have showed up one day.
Not much to say about snooze. It showed up in 1789 and is what’s known as a “cant word”, which basically means it’s lingo or jargon. It’s origin is unknown, and people think it might be echoic of the sound people make when snoring. Sure. Why not? Makes more sense than most of the other words today.
Nap comes from the Old English hnappian, to doze or sleep lightly. And that’s it. It’s origin is unknown. As is why it had an H. No relation to nap as in “downy coating” or the fuzzy ends of fibers or knap. Or knapsack, which is somehow not related to knap.
Tony Jebson’s page on the Origins of Old English