Italian miniatura originally denoted the painting of small images to decorate the initial letters of chapters in manuscripts (compare the use of post-classical Latin miniare in the sense 'to rubricate'). As these images were necessarily small, the term came to be used for small portraits, probably reinforced by an association by folk etymology with (ultimately classical Latin) min- in minore MINOR adj., etc., which has probably also affected the development of the extended senses in English and in other languages.
So mini and miniature are unrelated to minor, minus, minuscule and minimum. The latter four words are from Proto-Indo-European *mei- "small".
*mei- also gives us meiosis from Greek μείωσις "lessening".
And minister from Latin minister "servant, subordinate" - as in "inferior". The Latin minister is also found in minestrone - from Italian minestra "dish" plus the -one suffix. Minestra is from Latin minestrāre "to provide, supply" from minister.