The question has to lead to a risky thesis for a good essay and be unanswered in the book...a question that will lead to a good discussion and where I will find supporting details in the book..
I think the biggest aspect of this novel that stands out to me is the whole concept of duality and the existence of a good and evil side within us all. The question that haunted me after reading this novel was how to be integrate and accept both elements, rather than trying to isolate one, so that we can live a full and complete life. This is a key Gothic theme that is explored through this novel, and certainly owning up to the darkness, as well as the light, within us is a very important stage in terms of our own psycholgical maturity.
One of the big questions woven into Dr. Jekyll and Mr. Hyde is that of how to live a sincere, genuine life of integrity in your personality and actions. Watching Dr. Jekyll, it is apparent that it is not easy to stand your ground and find your self among the expectations of the world you live in. In this story, the expectations that Jekyll faces are to be the most noble and beneficent in his field. Today, an equally present pressure is to be as disruptive of an idealized goal as it is possible to be. So the question facing Jekyll is the same one that is faced today: How do I know my own mind and stand firm in my own integrity without being pulled by expectations that are external to me, to what I like, to what I want, to the value I want to contribute to the world and to my future. Today, even the question gets turn around so that a negetive Mr. Hyde answer of "I wnat to contribute chaos and destruction" is thought to be a valid option. But is it?
There are a lot of great discussion topics for this book, because it toys with the idea of good vs. evil, and the evil nature that dwells within all of us.
Here are some possibilities:
How does giving in to our vices (weaknesses, bad habits) turn us into a figurative Mr. Hyde?
Does everyone have an evil side, and if so, what is the best way to handle it?
Is the excitement and thrill of doing forbidden things worth the price you have to pay?
What are the pros of living a value-filled, calm, peaceful life that is within the bounds of societal laws and standards?
I hope that at least one of those strikes your fancy, or at least gives you a launching point! Good luck!