J was for Jasmine last year, and even though Jasmine is also one of the few color shades that start with a J, I'm not going to repeat myself. You can read about the Jasmine Princess here.
The second runner-up was Jazzberry Jam, but sadly that search did not produce any folktales. Go figure.
So that leaves us with Jade.
Jade is a pale green gemstone especially popular in Chinese and Japanese culture and mythology. The Jade Rabbit is a mythical creature that lives on the moon; the markings on the full moon's surface reminded the Chinese of a rabbit with a mortar and a pestle, and this is the story that came out of it.
The Jade Rabbit is the companion of Chang'e, the moon goddess. He is making the elixir of life in his mortar. The story doesn't exactly tell us how he got there; but we do learn about Chang'e.
Chang'e and her husband Houyi the Archer were immortals. But one day ten sons of the Jade Emperor turned into ten suns (get the pun there?) and threatened to scorch the earth. Houyi shot nine of them, which saved the earth, but ticked off the Jade Emperor, who banished the couple to live as mortals. In order to regain their immortality, Houyi went on a quest and acquired a pill of immortality from the Queen Mother of the West. While he was not home, however, Chang'e accidentally took the entire pill. You know what happens if you overdose on immortality? You become light and fly up to the sky, apparently. That's what happened to Chang'e and she has been living there ever since, with the Jade Rabbit to keep her company.
Fun fact: The Chinese moon rover Yutu (Jade Rabbit) landed on the Moon last year. The mission was aptly named Chang'e. Love it when old stories and new discoveries align.