Emily Grierson, like the previous answer correctly states, is someone who resists change. Unfortunately for her, change is everywhere around Miss Emily. All over town, in the fictional Jefferson, Yoknapatawpha County, Mississippi, society is slowly moving away from its old rule of social and racial separatism. Moreover, old, grandiose names, such as the Griersons', are starting to die out, losing with them all the former glory that would grant some families special considerations.
All this brings us to the issue with the mailbox. That Emily refused to add a mailbox and become a "regular" among the people comes as no surprise.
Emily grew up in a time and place where things were granted to some and denied to others. For example, when Emily's father died, then-mayor Colonel Satoris granted the Griersons an exemption from paying taxes. This was not something that the newer generation of town rulers would tolerate but, nevertheless, Emily still refused.
Emily also refused other things: She refused to give up the body of her father after he passed away for three days until she broke down in tears and was made to do it. She refused to do anything about the horrid smell coming out of her house when the town complained about it. She refused to let go of Homer Barron when her cousins came from Alabama to ask her to stop seeing him. She then refused everyone entry into her household until she finally came out one more time, for a short time, only to retreat back inside and never come out again until her death.
Therefore, as the previous answer states, Emily's stubborn nature is her standard practice. Add to this that Emily has feelings of entitlement that stem from her past as a member of a formerly grand family, and that she is entirely out of step with the changing times. What you will get is a difficult old woman who, despite of her age, never once grew up, or matured, into full adulthood. That is her very tragic flaw.