It is likely that the moon was discovered as soon as people decided to look to the sky. The moon is a lot older than the human race, so it would have always been in the sky for people to observe.
Even the dinosaurs used the moon in order to hunt. The moon has played a large part in the lives of humanity as a source of light in the night and ancient people have tried to understand the role of the moon in the grand scheme of things. Scientists today are still learning things about the moon, so it is possible new discoveries will be made.