And now for quiet words, because I like things quiet.
Quiet showed up in the fourteenth century, from the Old French quiete and classical Latin quies, rest, and its verb form quietare, to lull. Further back, it’s the Proto Indo European kweie-, to rest or be quiet. It’s remained remarkably consistent through the millennia!
Silent showed up in the sixteenth century, while silence showed up much earlier, in the thirteenth century. The former comes from the classical Latin silentem, from the verb silere, to silence, while silence was the Old French silence after being the Latin silentium, and obviously it’s also from silere. Again, very consistent.
Hush appeared in the mid sixteenth century, a variant on the Middle English huisht, and is thought to be imitative—in other words, the sound when people hush you sounds like hush, so it became a word. Shush is the same way, except it only appeared in 1921. Going “shhh!” to someone is older, having started in 1847, and shush comes from that. There’s no evidence that hush had any influence on it, but that wouldn’t surprise me.
Finally today, mute. It showed up in the late fourteenth century from the Old French muet, which is actually a diminutive form of mut/mo. Before that, it was the classical Latin mutus, which basically means speechless. It’s possibly from another imitative word, meue-. You might be asking how that’s initiative. Well, what sound do you make when you can’t open your mouth?
Tony Jebson’s page on the Origins of Old English