Just to put a little different spin on your question, and to consider the word "appreciated" to mean the sheer numbers of people who know and understand, I would say that Hamlet's "To be or not to be" speech to be the most appreciated part of Hamlet. There is so much literary criticism written on that one speech alone, I wonder if one could read all of it in a lifetime! Is it a contemplation of suicide? Is it something far more general? Deeper? The list is endless!
I can truthfully say that my own students, even the ones currently being taught American Literature, appreciate this speech simply by sitting in my classroom. Why? A long time ago, I got permission from the author of Calvin and Hobbes to reproduce and enlarge one of the cartoons that involves Calvin's dinner coming alive and reciting the speech. A true appreciation of that part of Hamlet? Hmmm, not so much, but a minor appreciation nonetheless.
One aspect of Shakespeare's Hamlet that is appreciated by some but cursed by others is its ambiguity. Scholars and others interested in analyzing literature often love the play's ambiguity, as do actors, who have so much freedom to interpret characters when performing the roles. Students trying to put together concrete ideas about the play, or those who like their tragedy to be tidy and neat, sometimes dislike the play.
So much is left up to interpretation in Hamlet. The question of why Hamlet waits has been argued by countless critics. One can find solid evidence for four or five interpretations, and traces of evidence for four or five more. And each interpretation reveals a different aspect of Hamlet's personality, and thus, the human condition.
The play is a figurative gold mine for someone who likes to break literature down and figure out how the play works, and how Shakespeare accomplishes what he does.
Besides, humans like to research and discover ideas and answers and solutions on our own. We like the feeling discovery brings. And, perhaps, no work offers as many opportunities for discovery as does Hamlet--due to its ambiguity and its openness to interpretation.
I would say the skill of critical thinking is the most appreciated part of the play. The very idea of Hamlet being a “thinking person’s hero” is something that is quite unique to the play and to literature of the time. Shakespeare did not create a swashbuckling hero, a warrior like Macbeth, or a regal king like Lear. Rather, he created an adolescent whose unique ability to think and continue to think is what makes Hamlet as both character and play so significant and so unique. Hamlet represents much of the modern condition where individuals possess the notion of thought and the understanding of pensive autonomy. Yet, while in distinct possession of these gifts, they are unable to do much with them as they find that their ability to think about situations does not allow them to transform them to their liking. This is what makes the drama so different than most others. Hamlet becomes the first intellectual thrust upon the stage of literature, and through him we also understand the pain in operating on such a stage without script, director, or stage directions. We can only act.