There are two questions that immediately arise. The first is why is this belief still held, including by many well educated people, in the face of a large body of evidence to the contrary? (and supplementary to that, why are studies to that end still being done?) Obviously cultural reinforcement plays a large role - the myth continues to be repeated as fact. The second is where did the belief come from in the first place? Is it some leftover animal instinct where it was easier to deliver and/or protect newborns by the light of the full moon? Is it some cultural remnant from days when we worshipped celestial events like full moons and solar eclipses; some long, lost supersition? Is it simple recall or confirmation bias among mothers and hospital staff?
As for other common lunar beliefs, such as increased psychiatric disturbance or more full moon emergency room visits, again the literature doesn't support those beliefs (with the exception of a single study that showed clustering of seizures around the full moon). The Skeptic's Dictionary has a summary of full moon myths and some debunking. The bottom line is that the full moon has little effect on birth rates, hospital emergencies, stock market prices and (despite the root of the word) lunacy. Unless, of course, you're a werewolf.