汉语「虹膜」译自英语 iris,而这个英语词又来自于希腊语,本义是「彩虹」。
至于为什么眼睛里的「虹膜」因「彩虹」而得名,Online Etymology Dictionary 上有解释:
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].
即,希腊语 iris 一词并不专指彩虹,而是可以指任何颜色鲜艳的圆形物体;而虹膜恰恰给了眼睛鲜艳的颜色。