I think the Blue Moon Azeroth has is Elune. More so, I think it's a celestial keeper with far more power than any other keeper. Or it is the full form of the Pillar of Creation, and the Tears of Elune is just it's conduit on the planet.
It harnesses power of Azeroths primary moon, and acts through it. When the Dragon Aspects were first created, the 2 moons lined up and formed "The Embrace" granting them their powers. This is also witnessed during the Blue Dragonflights selection of a new aspect. The Embrace formed and granted Kalecgos the powers of an aspect. I think the Titans placed the Blue Moon in the sky to watch over Azeroth from outer space, where it couldn't be corrupted, or attacked.
She has a connection to the Well of Eternity, when the Dark Trolls began to worship her, she taught them Titan words such as the name of the continent Kalimdor. Eonar also has a planet and sanctuary with a similar name. If Eonar created her to watch over Azeroth, then she would follow the same naming pattern.
I also think this serves as a beacon, a marker, for the Titans to locate the planet again. Agrammar went to primal Draenor, and helped seed life there, but had to leave. He planned on returning to the planet..Draenor also has a blue moon.
Any planet the Titans helped should have 2 moons in the sky.
i know everyone associates the White Moon with Elune, but I think it's just a moon, I mean after all, the Tears of Elune are blue, not white.