A good research question on almost any work of literature is this: why is this work considered an effective piece of literature? In other words, how does this work function as a successful piece of writing -- not as a successful expression of ideas (which could just as easily be communicated in an essay) but as a compelling, powerful, memorable use of language? A survey of different critics' opinions might be helpful in pursuing such a question.
You might want to examine how the dissonance of personality that is evident in the play was or was not present in Williams' own life. Did he suffer the same angst of an out-of-control personality as his characters suffer and seek an avenue of self-expression of the angst? Or did he wish to expose what he objectively saw as an evil of the milieu in which he lived and worked?
Would you be able to compare and contrast different interpretations of the play as presented by different casts in different locations at different times? How have the different stagings reflected contemporary issues and viewpoints while presenting the story of Blanche and Stanley?
I would advise you to pick something that you are interested in and that attracts you. Having read and watched the play, what interests you? What questions do you still have? You might want to focus on a character, such as the gradual breakdown of Blanche and the reasons for this, or Stanley's brute masculinity and how he interacts with others. You might want to pick a theme that is dominant in this book, such as fantasy vs. reality, or sexual violence. Whatever you go for, I think you need to pick it and you need to select something that you are interested in. Good luck!
A professor of English and author of several books on Streetcar, Philip Kolin offers this perspective:
"People have said that Williams absolutely invented the idea of desire for the 20th century....It was a play that dealt with for the very first time on the American stage, female sexuality and male sexuality."
This play replete with passion expresses much of the human condition.Williams used more than just the name of a streetcar for inspiration. In fact it has been suggested by some critics that the rough, ill-tempered, but rawly masculine Stanley was based on Williams' lover at the time he was writing the play, a man named Pancho Rodriguez Gonzalez.
Perhaps, then, you may wish to research how much of the play is based upon reality with the personal relationships and with the life in the New Orleans at the time.
You could examine the use of names in the play. I have always been intrigued by the name choices made by authors.
Another way to examine a text is through the lens of a critical theory. There are multiple critical theories which one can apply to a text. Here is a ling of the most used theories:
It depends on how much leeway you have with this assignment. When I saw your question, I immediately thought of Hurricane Katrina and the effect it has had on New Orleans. I can think of a lot of related research projects. You can compare the values and cultural aspects of New Orleans then and now. Here is some background:
Choosing a research question involves thinking about literary theory or criticism, for those disciplines provide intellectual frames for analysis of drama.
If you choose a Freudian or Lacanian approach, you might research a specific psychological trait of one of the characters. A Marxist approach might show how money and class affect the way characters think and interact with their environments. A feminist approach could investigate how the situation of Blanche is a result of systematic oppression of women. A New Historicist would look at the actual historical circumstances of the south in the period, and perhaps show how popular magazines could shed light on the play.
Once you have a preliminary theme consult the MLA International Bibliography to see how many articles have been written about your question. If you find over 20 articles, you need to narrow your question.