You already have a nice response but your request for help gave me a little more information (which I've added to your Question), so, based on that, I'll add to the discussion a bit.

Chekhov's "The Lament" is a tragic tale of an old, worn out Russian cabdriver, with his cabbie horse, whose beloved and cherished son has suddenly died in the hospital. It is coming up on a week after his death and Iona, the cabbie, has been unable to get anyone's attention or interest so that he can talk, to "tell it slowly and carefully," about his son's tragic death and his own tragic loss. In the process of trying to find anyone who will listen a moment and shed a tear for him and his lost son, "My son has died, and I am alive," Iona is abused and slighted and berated and struck by cold callous young men who ride in his cab. He laments for a woman's ear to tell his death lament to. In the end, he tells his tale to his beloved horse, which breathes on his hand and listens.

It is better, too, to talk to women; two words are enough to make them sob.
The little horse munches, listens and breathes over its master’s hand...

Now ... to your lament ... to your request for help with a sophisticated and complex addition to the "academic conversation." From the brief summary of the story, several areas of possible research choices emerge. One approach to combining English language and science with this story would be to use the sociolinguistic subject of forms of address in combination with the psychological subject of domination or of "bullying." Both subjects draw a lot of attention today so you can find research on both. In addition, with multiculturalism so much in the fore, forms of address should also yield research results, as will any discussion of the "other." In this story, domination of the other is from the higher social class to the lower social class within the same culture and society: the arrogant young male riders (which might be a significant research point right there: Why only male riders? Is it Chekhov's comment on young men or society's structure of excluding young women?) to Iona, the cabbie.

There are some specific passages to take note of in relation to the ideas above. One is when Iona is driving the young officers and "twists his mouth into a smile" so that, "with an effort," he can say that his son has died. This is interesting as sociolinguistics (English language) and as psychology (science). He, the other, must show servility and must placate the dominating factor by smiling even to say something so sad as "My son, Barin, died this week." You might ask: What gestures of address must be made to couch emotional expressions by the other so they are acceptable to the dominant ones? You might also ask: What psychological adaptation or maladaptation is required to be able to falsify facial expressions accompanying otherwise sincere and genuine communications of deep emotion that go contrary to the facial expression? 

Another passage to take note of is when Iona has returned to the stable and is wishing he had a woman to talk to. Neurobiologist, Louann Brizendine, M.D., has written the bestselling book The Female Brain that presents extensive research on the neurological composition of the female brain. This is a splendid avenue for incorporating cutting-edge science into your literary discussion. Linguist, Deborah Tannen, Ph.D., has written many books including the 1987 bestseller You Just Don't Understand that establishes the linguistic differences between men's communication perceptions and style and women's. This is an avenue for incorporating cutting-edge English language sociolinguistics into your literary discussion.

Finding just the right topic is one of the most common struggles with research in any subject area, and it is particularly troublesome to literature students because there are so many potential subjects on which to write. You mention English literature, so that does narrow the field a bit, but there are still so many choices.

First of all, it is important to select a work that you particularly like or that you find particularly interesting and would like to know more about. You will be spending a lot of time on this project, so it is always better to select something you will be able to maintain interest in. If you enjoy poetry, choose a poem; if you enjoy drama, choose a play; if you enjoy fiction, choose a short story or novel. Your selection must meet the guidelines of the assignment, of course, but it should also be something interesting to you.

In terms of what this kind of research entails, I have attached a great eNotes site on how to write an English literature essay. It shares something important about the kind of work you will be doing for this paper, at least based on your description of the assignment. In part it says:

You get [understanding] by reviewing the work closely and repeatedly, and by looking at content, form, and function. [E]xamine each choice made by the author until you can explain how it relates to the whole. Slow down, and take all elements of the work into account: sound, shape on the page, structure (chapter length, number of chapters, etc.), point of view, and so on.

Finally, the Purdue site I have attached will also give you some specific ideas about the kinds of topics which you will probably find helpful. Perhaps you can do a character study, such as comparing the sisters in Pride and Prejudice or examining Hamlet's indecision in Shakespeare's Hamlet. You might look at a text from an historical perspective, such as the conditions in England at the time of any of Dickens's writings, or how a specific time period impacted an author like Thomas Hardy. Another idea is to compare and contrast one work with another, one character to another, or one work to history. 

It's really difficult to suggest any specific titles for you, since so much of literature study is based on personal preferences and what you have and have not read; and your selection has to suit your professor's assignment and the length of the paper, as well. I would simply encourage you to think specifically about whatever work you choose. For example, rather than saying you are going to study Hamlet in Hamlet, you should focus on one aspect of his character and how it is displayed in the play, like his acting, his indecision, or his relationships with women. Again, choosing your topic is crucial, so consider what you like, what you've read, and what your exact assignment is before selecting anything.