late 14c. as the name of a flowering plant (Iris germanica); early 15c. in reference to the eye membrane, from Latin iris (plural irides) "iris of the eye; iris plant; rainbow," from Greek iris (genitive iridos) "a rainbow;" also "iris plant" and "iris of the eye," a word of uncertain origin, traditionally derived from PIE root *wei- "to bend, turn, twist."
Iris was the name of the minister and messenger of the Olympian gods (especially of Hera), visibly represented by the rainbow (which was regarded as the descent of a celestial messenger). From the oldest parts of the Iliad the word is used of both the messenger and the rainbow.
The eye region was so called (early 15c. in English) for being the part that gives color to the eye; the Greek word was used of any brightly colored circle, "as that round the eyes of a peacock's tail" [Liddell & Scott]. Another sense in Middle English was "prismatic rock crystal." Related: Iridian; iridine.