When asked at a seminar at the University of Virginia about the meaning of the title "A Rose for Emily" Faulkner replied, Oh, it's simply the poor woman had no life at all. Her father kept her more or less locked up and then she has a lover who was about to quit her, she has to murder him; it was just 'A Rose for Emily' ... In another interview, asked the same question, he replied, I pitied her and this was a salute, just as if you were to make a gesture, a salute, to anyone; to a woman you would hand a rose, as you would lift a cup of sake to a man. What do you make of Faulkner’s response? What else might the title suggest?

The two quotations express pity for Miss Emily, though the first is a little more opaque, and suggest that Faulkner's story was named as a kind of gentlemanly gift honoring the unfortunate (fictional) character. One thing that can make Faulkner's quotations a little hard to interpret is the fact that Faulkner was often frustrated with academic literary criticism and made statements in interviews simply to puzzle or have fun with his interlocutors. His excessive consumption of alcohol may also at times have affected his responses to interview questions. 
On one level, giving a rose to someone signifies love, but in a manner that is quite in keeping with the traditions of the Old South. It suggests that Miss Emily is a traditional southern belle and that the story will be a story of love, which it indeed is, albeit with a rather grotesque twist.
Rose imagery also shows up in the description of the bedroom at the end of the story with "the valance curtains of faded rose color, upon the rose-shaded lights." From this confluence of the title with the final rose imagery, we can assume that we are being asked to think about the contrast between traditional expectations for an upper class woman of the Old South and the reality of the New South. The rose-colored parts of the room—the traditional parts—are faded and worn; they remind us that Emily was part of a tradition that was rapidly fading itself, dying out to make way for progress. Emily, with her dead lover locked away in her home, clings to tradition despite its decay, as the faded rose-colored curtains suggest.