UNISEX:
1. BADR (بدر): Arabic unisex name meaning "full moon."
2. CHANDRA (চন্দ্র): Hindi unisex name meaning "moon." In mythology, this is the name of a lunar and fertility god.
3. MEZTLI: Nahuatl unisex name meaning "moon."
4.
YUE (月): Chinese unisex name meaning "moon."
MALE:
1. AKU: Babylonian myth name of a god of the moon, meaning "moon." Compare with another form of Aku.
2. BADRU: Egyptian name meaning "born during the full moon."
3. CHANDER: Variant spelling of Hindi Chandra, meaning "moon."
4. CHANDRAKANT (चन्द्रकान्त): Hindi name composed of the elements chandra "moon" and kanta "beloved," hence "moon lover." In mythology, this is the name of the moonstone that was created by moonlight.
5. IAH: Egyptian name meaning "moon." In mythology, this is the name of a god of the moon.
6. KORAY: Turkish name meaning "ember moon."
7. MARAMA: Polynesian myth name of a moon god, meaning "moon."
8. NANNA: Sumerian name meaning "illuminator." In mythology, this is the name of a god of the moon and son of Enlil. The crescent (or cup, receptacle) is his symbol. He is also known by the name Sin "wisdom." Compare with feminine Nanna.
9. PULAN: Chamoru name meaning "moon."
10. RAMACHANDER: Variant spelling of Hindi Ramachandra, meaning "Rama-moon."
11. RAMACHANDRA (रामचन्द्र): Hindi name composed of the name of the god Rama and the Sanskrit element candra "moon," hence "Rama-moon."
12. SIN: Sumerian name meaning "wisdom." In mythology, this is the name of a god of the moon and son of Enlil. The crescent (or cup, receptacle?) is his symbol. He is also known by the name Nanna ("illuminator").
http://www.20000-names.com/moon_names.htm
I'm REALLY fond of a Basenji named Sin. LOL
Aku and Chander are my other votes.