What does a blue moon in your eyes mean?

The writer is showing that the person is "one in a million", "very special", a la "once in a blue moon", not an every day occurance.

Read the lyrics prior to this lyric and it should open up for you.

"Momma said you were the chosen one" and "You have to burn to shine"...then "born under a bad sign with a blue moon in your eyes"...

I would say unlucky to be this rare figure in this story....