Since I've recently talked about Rainbow Brite and rainbow-related classical mythology, I pretty much feel obligated to discuss Polychrome, the Daughter of the Rainbow in the Oz series. I've mentioned her a few times, and of course she's in one of my icons, but I don't think I've done an entire post about her yet. bethje is always comparing her to Rainbow Brite, but while the cartoon character had an important job to do, Polychrome and her sisters really don't seem to have much to do beyond dancing. Still, Polly (as her friends call her) definitely increases in competence throughout the books, even if most of her actual growth is offstage. When she first shows up in The Road to Oz, she's presented as a beautiful and fragile creature totally out of her element. She accidentally danced off the end of the rainbow onto the ground, and she soon meets up with Dorothy's Oz-bound party. She's constantly in motion, and prefers to eat ethereal foods like dewdrops and mist-cakes. When she tries human food, she enjoys it, but eats only a very tiny bit ("about as much as a fly would eat," according to Dorothy). Polychrome dresses in robes that shine with all the colors of the rainbow, and while this is never specified in L. Frank Baum's text, John R. Neill consistently draws her with a skullcap.
When Polychrome returns for a brief appearance in Sky Island, she's hardly helpless, and instead assists the earthly visitors by arguing law with the Pinkies. Maybe her increased confidence has something to do with her being in her element (i.e., the sky), but it also seems like she's gained more practical knowledge.
The rainbow fairy's next role is in Tik-Tok, which reuses the plot device of her being stuck on the ground when the rainbow fades away. Oddly enough, when she meets the Shaggy Man, one of her companions in Road, she doesn't seem to recognize him at all. There's been much speculation on this point, with the generally accepted Oz-as-literature explanation being that this was a result of Baum's carelessness in adapting his play The Tik-Tok Man of Oz (which wasn't entirely consistent with the books, although it used many of the same characters) into novel form. Perhaps the best Oz-as-history explanation is that Polychrome was too upset to recognize Shaggy at first, but March Laumer's Careless Kangaroo contains a much more convoluted explanation about how she was in love with Shaggy, and decided to have her memories of him wiped out when it was obvious he didn't return her affection. This book also has her studying at the Wogglebug's college, which explains her increased intelligence in books after Road. An interesting aspect to Tik-Tok is that the Nome King seems to have a crush on Polychrome, although it's possible that he's bluffing.
The Daughter of the Rainbow shows up again in Tin Woodman, in which she's caught by the giantess Mrs. Yoop and transformed into a canary, and the Tin Woodman's party has to help her escape from Yoop Castle. Despite her foolishness in being caught, Polychrome is quite competent during this adventure, even while in canary form. She performs magic on several occasions, which is definitely a change from how she claimed not to know any magic back in Road.
The later canonical authors never gave Polychrome another prominent role, but they did seem to like her anyway, having her make cameo appearances from time to time. She assists characters in using the rainbow itself to reach Oz in Ruth Plumly Thompson's Grampa and Purple Prince, as well as Neill's Lucky Bucky. She and Kabumpo recognize each other in Purple Prince, and Wonder City has an episode in which Polly and Number Nine know each other from playing together on rainy days, so it's pretty much inevitable that the fairy made other unrecorded visits to Oz. I've seen rumors that Polly was a major character in Jack Snow's unpublished manuscript Over the Rainbow to Oz, but since this story has yet to turn up, there's no confirming that.
What exactly being "Daughter of the Rainbow" entails is never clearly specified. Polychrome refers several times to the rainbow itself as her father, but doesn't mention a mother. Both Marcus Mebes' Lurline and the White Ravens and a story in the most recent Oziana link the colorful fairy to Greek mythology by identifying her mother as Iris. Her uncle, as confirmed several times, is the Rain King. Gina Wickwar's contest-winning Hidden Prince features appearances by Polychrome and her uncle, as well as Paddy O'Paint, the leprechaun in charge of touching up the rainbow. Her also mentions another leprechaun named Kelly, who keeps the pot of gold at the rainbow's end.