I feel the same way Dozus. Being a Man of The Beard myself they (beard-inclusive names) have a special feeling attached to them. I imagine myself with a glorious length of facial awesomeness worthy of being used to identify myself. When you are named for your beard, you know you have a great beard.
My second choice was MoonHunter, a Hunter under the light of the Moon, presumably hunting creatures of the night with wanton badassery.
Moonlake is nice, simple, clean with some powerful imagery attached, but it doesn't move me in the way I think the "Best" name should.
MysticMoon sounds like the name that would be given to a magical order. Mages of the Mystic Moon or similar. Certainly thought provoking, but not the best of the batch.
Moon-Child is probably my least favorite. It feels overdone. It might have some significance to the owner, but there are likely more owner's who just thought it sounded cool.
Go, Moonbeard! Best "Moon" Name OF ALL TIME!