In Book 1 of Valerius Flaccus' Argonautica (which is by a Roman writer pertaining to Latinised Greek mythology), the Argo gets caught in a violent storm on the Aegean, which is summarily calmed by the sea-god Neptunus (= Poseidon), and thereafter the sky clears to a sunny day, complete with pretty rainbow as the clouds rise further and further away from the ship.
The goddess Iris does feature in quite a few myths/stories, so technically each of those features the rainbow, especially those in which the goddess travels from one place to another, because the rainbow was supposed to be the wake (trail) of her colourful swathes (robes) as she flew through the air. Homer says in the Iliad, though, that the rainbow was a portent sent by Zeus to foretell war and storms.
Aaron Atsma, the author of The Theoi Project website, says that "The Lyric poets made Zephyros the father of Pothos (Passion) by Iris the rainbow. The imagery of rainbow (iris) and west wind (zephyros) symbolised the variegated brilliance of passion (pothos)." Iris had a twin sister named Arke, who, according to Atsma, "was the messenger of the Titan gods ... She may have been associated with the faded second rainbow sometimes seen in the shadow of the first. During the Titan-Wars the two goddesses served on opposite sides—where Iris became the messenger of the Olympian Gods, Arke assumed the role of messenger for the Titanes. At the end of the war, Zeus stripped her of her wings, and cast her into the Tartarean pit along with her masters." Incidentally this story is supposed to explain how Achilles, the greatest Greek warrior of the Trojan War, was the fastest runner in the world in his time. Arke's wings, after they were severed, were millennia later given as a wedding-gift to the sea-goddess Thetis. The Centaur Kheiron [Chiron] later attached these wings to the ankles of Thetis' son Achilles.