The concept that would draw me first is that of honesty as a philosophical concept.
What does it mean to be honest? If you believe your actions to be moral and driven by harmless impulses, does that mean your actions are moral and harmless? Does that mean that you can lie and be honest at the same time, as Gatsby seems to believe as he induces a married woman to elope with him in the name of "true love"?
Nick is another interesting character when looked at in light of questions of honesty. If he believes what he says, as inaccurate as his views may be, does he maintain an honesty through ignorance? If he conveys a larger truth, so to speak, while telling small lies in his narrative, can we call the narrative true? Can we call it honest?
Is there a direct correlation between honesty and truth?
You could consider reading or reviewing the novel through a specific philosophical lens such as Marxism. Even though Fitzgerald probably had no intended Marxist message, we can still study what Fitzgerald's novel reveals about the class system. What is the dynamic between the proletariat and the bourgeoisie? What does status in society give a person? How do the classes act? How do economic factors influence behavior?
How about taking off from what T. S. Eliot called the "mythic method"?
This "mythic method" is what T.S. Eliot expounded upon in his discussion of James Joyce's Ulysses:
"In using the myth, in manipulating a continuous parallel between contemporaneity and antiquity, Mr. Joyce is pursuing a method which others must pursue after him ... It is simply a way of controlling, of ordering, of giving a shape and a significance to the immense panorama of futility and anarchy which is contemporary history."
In The Great Gatsby, there are, in fact several mythical allusions: Gatsby is referred to by Nick as Trimalchio, and Gatsby's car has "fenders like wings" with "triumphant" hat boxes and a "labyrinth of windshields that mirrored a dozen suns." And, it does seem that Fitzgerald gives "shape and significance" to Gatsby's futility.
Excellent ideas so far. I'm also thinking of self-delusion as a pervasive theme in the novel. First Gatsby is convinced Daisy will love him if he can make enough money; then he is convinced she loves him enough to leave Tom, has always loved him, and never loved Tom. These all turn out to be self-delusions. The very persona of Jay Gatsby is a grand self-delusion, as well. Nick tells us that he is the only honest person he knows, yet we know that is an overstated and certainly untrue assessment. Perhaps in the world of East and West Egg Nick is the most honest person he knows; however, that is clearly a self-delusion. In terms of philosophy, then, what causes one to embrace such delusions even in the face of stark reality would make an interesting study.
The Great Gatsby offers many good topics for you to consider. One that has always interested me is the conflict between illusion and reality. Can illusion be maintained in the face of reality? If so, for how long and at what cost?
Gatsby dedicated his life to achieving what he believed was possible, a romantic dream of reuniting with Daisy and repeating their past. When Nick tells him that nobody can repeat the past, Gatsby was shocked to even hear that suggestion. He had lived with an illusion for so long it had become his reality. Gatsby's eventual destruction makes it pretty clear how Fitzgerald viewed the contest between illusion and reality, suggesting that reality will prevail, regardless of how hard one works to maintain an illusion. Also, George and Myrtle Wilson struggle to maintain their own illusions, with no better result. It is interesting that Gatsby and the Wilsons are the only characters who die in the novel.
An examination of the power of illusion in relation to reality might be good topic for you to explore. Many passages in the novel address it; finding evidence to support your discussion would not be difficult. Good luck!
I think that you can take this particular task in many different directions. One potential avenue might involve assessing whether or not there is a sense of truth and justice in the world today. One of the most stunning aspects of the novel is that the people who are the very worst in character such as Jordan and Tom really receive no punishment or deliverance for their actions. They willfully hurt others and almost take a sort of perverse satisfaction in the pain they cause others and how their own lives benefit from someone else's pain. Yet, people such as Gatsby, who might be guilty of many things but not deliberate cruelty, suffer the worst of fates. I think that the philosophical issue here is what defines punishment and what defines justice in this setting? There is a valid assertion here about the nature of justice and fairness and one that I think a great deal of philosophy can emerge. In this light, the question of justice becomes part of the novel's themes and part of its philosophical point of analysis.
Another slightly offbeat theme is that of dissatisfaction within the novel. To get 'philosophical' you might ask not only how characters seem dissatisfied on a physical/emotional level but also how dissatisfaction permeates Fitzgerald's approach and influences the shape of the story - you might also posit the question 'what does it mean to be dissatisfied' - for characters such as George, Gatsby and Myrtle this may seem obvious, however what is the root of the less specific dissatisfaction of Nick, Tom and Jordan?