A myth must have a god or goddess involved and explain a natural phenomenon that the ancients had a hard time explaining, such as the seasons. In this case, Artemis, goddess of the wilderness and hunt, is associated with the moon, with a crescent above her head. Her brother is Apollo, the god of the sun. They are very close. Artemis was very attractive and had many suitors, but she is the goddess of chastity and virginity. Half of the moon is always lit by the sun, so you could show that her brother always gave her love and warmth. He was protective of her. The moon phases are caused by the changing angles of the earth in relation to the moon. Create a story in which Artemis is in love with a human on earth and her brother does not approve. She is a goddess and must understand her duty to her followers so she can never be with her love. Her brother comes to visit her once a month. During his visit, we experience the new moon, because she blocks his sight of the earth. When he leaves and she can flirt with her love without fear of repercussions of her brother, we have the full moon. Her warmth shines down on earth and her lover, even though she can never reach him. I hope this helps. It will need to have details added, but it will help you get started.
The moon was once just a large asteroid. Then it crash into the top of the Earth, distroying itself and also some of the Earth. Pieces of the Earth and the asteroid fused together, and began to orbit the Earth. And now we call it our moon.