The moon has been implicated in changes in human behavior. Here is an article about a study about how the lunar cycle may affect sleep. Researchers studied volunteers that slept two nights in a completely dark bedroom where they could not see the moon. The results showed that around the full Moon, brain activity related to deep sleep dropped by nearly a third. Melatonin levels also dipped. The volunteers also took five minutes longer to fall asleep and slept for 20 minutes less when there was a full Moon. I don't have access to the actual article, so I am not sure if the volunteers subjectively reported that they felt their sleep was different. And in my experience, most of my patients with insomnia do not report any association with lunar cycle and sleep quality. Of course, women sometimes have subjective changes in sleep quality due to menstrual cycle, but that is not correlated with lunar cycle.
It would be interesting to see how the same experiment done over the entire lunar cycle.