We now know that the moon does not produce its own light, but instead just reflects the sun's light. How early was this understood? How was it figured out? Do any societies have myths involving this relationship?
Among the Greeks, Parmenides and Anaxagoras both stated this. Anaxagoras, 5th century BC: "It is the sun that puts brightness into the moon." http://demonax.info/doku.php?id=text:anaxagoras_fragments
For the earliest known Chinese statement of this, Needham (Science and Civilisation in China, vol 3, pg 227), quotes the Chou Pei: "The sun gives to the moon her appearance, so the moonlight shines brightly forth". The date that this text was written is unknown; Needham suggests the 6th to 4th century BC. There are later, well-dated sources from the late 1st century BC stating this.
The oldest surviving Indian statement of this that I know of is by Aryabhata, in his Āryabhaṭīya, from the end of the 5th century AD, where he writes that the moon (and the Earth, and the planets) are spheres illuminated by the sun (pg 64 in the Clark translation).
The idea follows readily from the phases of the moon, and provides a simple geometric explanation of lunar eclipses.