I am reading all of Shakespeare's plays in the course of a year; this is the first.
Why Hamlet endures: death
Death is the universal reality, and Hamlet is consumed with it. Hamlet meditates on death as annihilation, on the mysteries of the afterlife, and on the physical aftermath of the body. Laertes shows concern for the ceremonial rites to commemorate death. There is discussion of the religious/moral impact of death, about whether characters are able to be absolved of sins before death. Death is the universal theme, and so Hamlet will always have universal appeal.
Why Hamlet endures: language
The plot is familiar, and the emotional impact may be lost, but the language endures and will endure. This is just spectacular poetry, indescribably great. The wordplay is genius, and the poetry awesome. Even trying to find words to express the beauty and brilliance of Hamlet's language seems cheap.
Why Hamlet endures: pretentiousness
Most people who read or watch Shakespeare these days are intelligent, and probably self-aware of intelligence. And so it would not surprise me if most readers of Shakespeare have at some point thought, consciously or unconsciously, something like the following:
"I'm very, very smart. It's my lot to be surrounded by small-minded fools: some of them have positions of authority over me, and many of them are out to get me. But I have more wit and insight than any of them: I'm smarter than them all."
Such readers find a character to relate to in Hamlet.
If Shakespeare ever set out to write a play to say "Revenge is folly. Violence is folly. Vengeance is cyclical, and only brings about more bloodshed. All violence, whether evil or 'righteous,' can only bring horror," he would have succeeded in Hamlet. Of course I don't think Shakespeare set out to write Hamlet with such a precise goal, but the play still speaks to me as a pacifist.
I used to see a conventional catharsis in the ending of Hamlet: the country was sick, there was a great purging, and now Fortinbras comes to restore order and health. Then I saw the '06 Hamlet at the Guthrie in Minneapolis: Fortinbras is portrayed as a preening, ambitious warmonger, and ascension to the throne is a continuation of corrupt and brutal power. Now when I return to the text, I don't see how I missed it. Throughout the play, Fortinbras is only referred to for his belligerent nature and behavior. And at the end, when examining the carnage at Elsinore, he can only speak in terms of warfare, and he can only honor the deaths with the trappings of warfare. Whatever sickness was purged in Denmark, Fortinbras can only be expected to bring carnage.