Consider this: What is “mini” short for?
You could argue that it’s short for “minuscule” or “minimum.”
That would mean it’s related to “minus” and “minor,” which are direct adoptions from Latin, in which minus and minor mean “small” or “less.”
But in most uses, “mini” is short for “miniature.”
A mini dog breed, or a miniskirt, or a mini car, or even, in a way, Minnie Mouse—these all imply “miniature” things.
But here’s the curious part: “Miniature” isn’t etymologically related to “miniscule,” “minimum,” “minus” or “minor.” (Though “minister” is related to the latter four. † )
Indeed, “miniature” and thus “mini”—at least etymologically—have nothing to do with size, but with color.
“Miniature” is from the unrelated Latin miniare, meaning “to paint red,” from minium, the name of a lead-based red paint popularly used to illuminate medieval manuscripts.
A “miniature” was originally a word for one of these manuscript illuminations—especially (but not exclusively) one pigmented with minium.

The meaning of “miniature” evolved because:
1) The paintings in illuminated manuscripts were generally quite small, and
2) although it’s still in use today, a broader range of available pigments reduced the reliance on minium in illuminations, and thus
3) by influence of folk etymology, the reduced synonymy between minium and illuminations, combined with the similar structure of “minimum” and “minuscule,” shifted the definition and assumed origin of “miniature” to today’s meaning.
One critical stepping stone that solidified the modern definition was the portrait miniature, a portrait style popularized in the 16th century that drew upon similar techniques as mansuscript miniatures—but didn’t necessarily make use of minium. These portraits were similarly small in size to facilitate exchanges of the artwork in courtship, between family members and otherwise as gifts between the elites who could afford to have them produced.
† In addition to its Modern English meaning, “minister” is also a Latin word for an inferior ranking person, a servant or a priest’s assistant. The word extended into English with the implication of a servant of the people—as in a prime minister or administrator—or a servant of the church.