An Oxford University Press webpage explains:
It only makes sense that this word miniature would derive from the Latin word minimum, meaning “the smallest.” It only makes sense, but it’s wrong.Lots more at Wikipedia.
Miniature is one of those strange words that has an etymology that defies logic. The actual truth is that before things that were tiny were called miniature, a certain kind of small portrait was called a miniature.
Before that, the art of illuminating those beautiful letters and figures in hand reproduced .
ancient books was called miniaire in Italian.
This miniaire art was in turn named for the red color that was especially popular for use in producing this art.
The red color was usually produced by use of a red kind of lead and it was the Latin name of this red lead that gave the color its name because the lead was called minium.
Thus etymologically, miniature and minimum actually don’t even have a small relationship with each other.
The word miniature, derived from the Latin minium, red lead, is a small illustration used to decorate an ancient or medieval illuminated manuscript; the simple illustrations of the early codices having been miniated or delineated with that pigment. The generally small scale of the medieval pictures has led secondly to an etymological confusion of the term with minuteness...