We use hush, besides for talking about a great Buffy episode, to speak of a silence. It can also be used as an imperative to ask somebody to be quiet, though we are much more likely to use shush for that. We’ll talk these commands today.
These interjections are almost always onomatopeic in origin. That means they are words spawned from a sound which they evoke. So what sound is usually used to silence people? It needs to be imposing, to be able to rise over already present sounds in the environment. Since sibilant sounds fit that bill pretty well, it’s not surprising we find them consistently as the origin of these expressions.
Sibilants are made channeling air with the tongue so it flows to escape between the teeth: in shush, you can find a sibilant <ʃ> sound at the beginning and end of the word. We also find other sibilants in French chut!, Spanish ¡chitón!, Finnish hys! or Italian zitto!.
Now for the twist: one would expect (and I did, and that’s where this entry is coming from), that shut up would come from that same line of sibilant-silencing people, and that only later would expand its meaning to shutting other things like doors.
And I was wrong, since, in hindsight, it was a bit of a wishful assumption. The “sh” in shut up is but a coincidence, an evolution from Proto-Germanic *skut- (“to project”, like shoot) through Old English scyttan (“to fasten a door”).
Sheesh, what a shame! I shouldn’t share such shows of my short-sightedness.
Sources:       
Originally published in The blind mouse.