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.
I think one of the most significant purposes of the eyes is where they are located: over the Valley of Ashes.
Think about what happens here... they are all things that God sees that the rest of the characters may not see. For example, Tom picks up his mistress regularly here. He befriends her husband and right in front of him takes his wife away... regularly. Likewise, later in the novel, a great accident occurs here. The one who commits the crime is unknown and the one who is "punished" for the crime did not actually commit it. (But he certainly committed plenty of crimes) It is ironic that Wilson is the one who notes that God's watching because of what he actually does about it. I don't know where you are in the book, and I am assuming it's early so I am trying to be vague here.
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.
Of course, this is open to interpretation.
George Wilson seems to think that the eyes represent God. He seems to think that the eyes show that God is watching them, seeing what they are doing. We see this idea that we are always being watched in the character of Owl Eyes as well. They seem to be watching the people, implying that they have no moral values.
But Fitzgerald himself never actually says this clearly. He seems to be saying that the eyes only have whatever meaning the people in the book give them on their own.