I have taught minuets to scads of students--far too many to count, but today, for the first time, one of my students actually asked me why these things are called minuets. The Question of why a minuet was called a minuet never dawned on me, so I looked it up.
The name comes from the small, dainty step (mini steps, if you will). You can see those mini steps here. I also learned that "menuetto" is not a word, even though Beethoven used it once in a while. While I was at it, I looked up "gavotte." The name of this dance comes from Pays de Gap where the inhabitants were known as "gavots." Pays de Gap was located in the former province of Dauphiné, where, it seems, skiing has taken the place of dancing.