In the Poetics Aristotle describes the ideal tragedy as having a cathartic effect based on fear and pity. An interesting essay would illustrate how these emotions of fear and pity are evoked in the audience through one or more of the characters, most notably Hamlet, and whether their effect is to produce a true catharsis or rather despair. Also, if you use the Aristotelian definition of tragedy, consider whether this play measures up to his standards (e.g. if despair is evoked, can this play be a true Aristotelian tragedy?).
Another essay topic would be what exactly the tragedy of Hamlet was. Most people present as the keynote of the tragedy the appearance of growing insanity in Hamlet which results in his death and the death of many others. You could argue, however, that although this was indeed tragic, it was not merely Hamlet's decline which defined the play as a tragedy; rather, the tragedy was the decline in the social order--the decline in trust, confidence, truth, etc., which perhaps caused Hamlet's deterioration and which at the same time unfolds as he deteriorates.