Before talking specifically about Mangan's sister in James Joyce's "Araby," it's worth mentioning that Irish literature has a long history of using female characters as symbols and personifications of Ireland. One of the most famous examples of this trope can be found in W.B. Yeats and Lady Gregory's play, Cathleen ni Houlihan. In this play, the main character (the eponymous Cathleen) begins as an old woman wandering the countryside and lamenting the loss of her four green fields. This character is often read as a symbol of occupied Ireland, as Cathleen's four green fields roughly correspond to Ireland's traditional four provinces. Furthermore, Cathleen's transformation at the end of a play into a young woman can be seen as a symbol of Ireland's projected rebirth, one that occurs once she regains her sovereignty from Great Britain.
Let's consider this trope in conjunction with "Araby." In Cathleen ni Houlihan, the character Michael Gillane becomes infatuated with the old woman (who represents Ireland), and this infatuation drives him to join a band of Irish rebels fighting for freedom. In "Araby," the main character is similarly obsessed with Mangan's sister, and the thought of her is enough to rouse him to some pretty dramatic emotion: "her name was like a summons to all my foolish blood" (30). Like Michael, the unnamed narrator of "Araby" is infatuated with a girl, and, also like Michael, this infatuation drives the narrator to perform deeds to win over said girl. The difference is that, instead of heading off to war, the narrator goes to Araby in an attempt to buy Mangan's sister a trinket. Through these parallels with the classic personification of Ireland as a female character, we can see Mangan's sister as potentially symbolic of the island itself.
If we take this symbolism to be the case, then the end of the short story becomes very interesting indeed. At the end, the narrator fails to buy anything at the bazaar, and he realizes the foolishness of his actions and obsession. As such, it would appear that Joyce is throwing an element of disillusionment over Ireland's classic symbolic form. Perhaps, Joyce seems to be saying, it's not wise to allow metaphorical infatuation to govern our lives and drive us to perform deeds to prove our love/patriotism. In this sense, Joyce takes a step toward dismantling the symbolic female as Ireland trope.