The quotation has to be the title of the story but does not have to be anything to do with the actual play. It would also help if the quotation I use is well known.
What a great assignment. Given the magnitude of Shakespeare's work, there are multiple options one could use to create a short story using the title of one of Shakespeare's plays.
1. "All's Well That Ends Well"- This is a classic problem play. Therefore, a short story fits well using the title given the idea of completion and resolve. This title allows for the typical plot line (introduction, rising action, climax, falling action, and conclusion) will support the title itself. When writing a short story for this title, examine a subject which embodies either internal or external conflict which the protagonist questions their ability to come out of the story a new and different person. The protagonist needs to be dynamic (changes over the course of the action) and feel, at one point, that there is no hope. In the end, the character will find that the conclusion in life is all that matters: all's well that ends well.
2. "As You Like It"- This story could examine a character who is very weak and gives into all conflict around him/her. The title suggests weakness and the inability to stand up for one's self. Create a character and story line where the protagonist is walked over by all other characters. Irony could be placed in during the falling action or conclusion where the protagonist suddenly changes--the protagonist can (finally) state "as you like it" denoting his/her realization that he/she no longer cares and is not willing to bend over backward regarding the demands of others.
3. "Much Ado About Nothing"- This story could examine the problems associated with make too much of a situation. A writer could take something very little and turn it into something much more complicated than it should have been.
4. Any of the histories or tragedies- Given all of these plays use names as the title, it would be very easy to create a character based upon the name given to the play. Be sure to put a twist on the story so as not to repeat the idea and action embedded in the play itself.
Neat assignment. A classic, oft quoted line from Hamlet is from his 3rd major soliloquy: "To be or not to be, that is the question." Hamlet is talking about life and death, but more importantly, he is moreover talking about action and inaction, and that could made into a compelling short story topic.
Another favorite quote of mine, again from Hamlet is in Act II when Hamlet is talking to Rosencrantz and Guildenstern. He makes a crazy claim that Denmark is a prison and his friends disagree with him. Hamlet's response is: "there is nothing either good or bad but thinking makes it so." He means that however a person thinks about something, of feels about something, is how it IS for the person. I use a silly example like broccoli. If you don't like it, then it is bad. If you like it, then it is good. It isn't just one or the other -- it is whatever YOU think it is. This applies to almost everything! It is an interesting point on the power of personal observation, drawing personal conclusions, and having control over personal reactions.