It would help if you were more specific about the assignment. When in doubt, choose a theme and connect it to the author's style. You can look at specific literary devices, plot structure or just about anything. In this story, I might focus on symbolism. Then you can write about how symbolism is used to develop your theme.
If you are writing a literary analysis, you should first take a look at the themes you find in this story. Start by asking yourself, "what is this story about?". Generate a few topics from there, such as
- love (parental or courtship)
- loss of loved ones
- the way that society influences people's behavior
- the loss of social standing and/or reputation
From there you can derive your themes. For example,
- When love does not match a person's expectations, it can make them act irrationally.
Now you have to prove it! How does love make Miss Emily act irrationally? What literary elements does the author use to prove this?
- The town does not love Miss Emily anymore now that she has lost her money.
- Miss Emily's father loved her TOO MUCH, and never let her experience a love of her own.
- Homer stops loving Emily altogether.
These are achieved through things like
- setting: in the South, reputation was key--without reputation, Miss Emily had nothing.
- characterization: Miss Emily and Homer are complete opposites. He does not/cannot love her as she needs. Her father loves her too much, and is still a presence in her life even after he dies!
- foreshadowing: As Miss Emily is constantly "unloved" by people and the town, she makes small decisions eventually leading up to the horrific ending.
Now you have to prove it! Find evidence from the text to support each of your claims that PROVE whatever you are arguing--in this case, the argument is that when people's expectations of love are not met, it can make them act irrationally. You are arguing your theme, essentially, and you are proving it via the text.
To vary the argument of Emily's isolation as a cause for her aberrant actions as suggested above, the writer can also take the approach of "A Rose for Emily" being a tragic tale. For, Emily suffers from the weight of her name--Grierson. Tracing so many of her reactions and actions as well as the perceptions and reactions of the townspeople, the writer can make an argument for this perspective as well.
Another argumentative path that can be taken is the analysis of the so very clever and artistic employment of time that Faulkner has in this story. With his flashbacks and shifts in time, Faulkner's skillful use of this part of the short story element of setting prevents readers from "putting all the pieces together," thus helping to create the Gothic horror of the discovery in the conclusion. "A Rose for Emily" is divided into five sections, with the first and last sections framing the others in the present time while the three middle sections detail the past. Thus, the story begins and ends with the death of Miss Emily Grierson.
Of particular note is the fourth section, in which Faulkner writes of Emily,
Thus she passed from generation to generation--dear, inescapable, impervious, tranquil, and perverse.
Most interestingly, these five adjectives describe each section of "A Rose for Emily." Critics argue that these descriptions of the times of each section are a metaphorical characterization of the differing states of mind that the townspeople pass through in their evaluation of Emily. Certainly, there is no question that the shifts in time of Faulkner's "A Rose for Emily" serve to develop the horror of the story as well as explicate the narrative; moreover, they are essential to an understanding of this marvelous story.
In "A Rose for Emily," one could argue that forced isolation can lead to insanity. Emily's father forced her to live an isolated life. He would not approve of any of Emily's gentleman callers. He would run Emily's suitors away. No one was good enough for his Emily:
He [the narrator] also notes that Emily is apparently a spinster because of her father’s insistence that 'none of the young men were good enough' for her.
In turn, Emily was forced to live a life of isolation. She became more and more withdrawn as the years passed by. Ultimately, Emily lost her sanity along the way. She proved this by poisoning Homer Barron and then keeping his corpse to rot on the bed in which she slept.
Isolation can have extremely negative effects on a person. The loneliness can become unbearable. Obviously, Emily had her reasoning at one point in her life. The neighbors trusted her to give their children art lessons "for a period of six or seven years when she [gave] china-painting lessons."
Somewhere along the way, Emily lost her sanity. She poisoned Homer rather than spending her life alone any longer:
Soon after Emily is buried, several of the men force the upstairs open. There they find what is evidently the rotten corpse of Homer Barron. Even more grotesque, they find a long strand of iron-gray hair on the pillow next to his remains.