In the book The Great Gatsby, what function is served by the image of "the eyes of Doctor T.J. Eckleburg" brooding over the valley.

The novel The Great Gatsby is centered around illusion, or blindness.  The novel is the story of Gatsby trying to recapture the past, trying to recapture the love he and Daisy previously had.  The trouble is, the past Gatsby is trying to recover never really existed--it's an illusion.  Daisy never loved Gatsby the way Gatsby loves Daisy.  So the eyes of the eye doctor reflect the blindness of Gatsby.

Of course, numerous other examples of blindness exist in the novel as well.  Tom and Daisy are blind to the hurt they cause with their affairs, Wilson is blind to his wife having an affair with Tom, Nick is blind to his judgmental nature.   

There's more symbolism in the eyes, but I'll let other editors handle that.