1580s, "a reduced image, anything represented on a greatly reduced scale," especially a painting of very small dimensions, from Italian miniatura "manuscript illumination or small picture," from past participle of miniare "to illuminate a manuscript," from Latin miniare "to paint red," from minium "red lead," used in ancient times to make red ink, a word said to be of Iberian origin. Sense development is because pictures in medieval manuscripts were small, but no doubt there was influence as well from the similar-sounding Latin words that express smallness: minor, minimus, minutus, etc.
word-forming element meaning "one who does or makes," also used to indicate adherence to a certain doctrine or custom, from French -iste and directly from Latin -ista (source also of Spanish, Portuguese, Italian -ista), from Greek agent-noun ending -istes, which is from -is-, ending of the stem of verbs in -izein, + agential suffix -tes.
Variant -ister (as in chorister, barrister) is from Old French -istre, on false analogy of ministre. Variant -ista is from Spanish, popularized in American English 1970s by names of Latin-American revolutionary movements.