Atticus expects, after he dies, for Scout to get the pearl necklace that belonged to her mother, while Jem will get Atticus's watch. Scout being destined for the pearl necklace foreshadows that, despite her current tomboy status, Scout is destined to grow up to assume a traditional woman's role. She is meant to become a southern lady, like it or not. The pearl necklace symbolizes that gracious and restricted role and becomes a synecdoche.
A synedoche is a literary device in which a part represents a whole. A common example would be to use the word "crown" to symbolize the monarchy. In this case "pearl necklace" stands in for a everything it means to be a southern lady: this is underscored by another reference to pearl necklaces, when Scout mentions that
Aunt Alexandra’s vision of me involved playing with small stoves, tea sets, and wearing the Add-A-Pearl necklace she gave me when I was born.
Scout may be enjoying the somewhat anarchic freedom of childhood, but the novel suggests that is a brief interlude before she takes on her predefined woman's role.
Scout will receive the pearl necklace and a ring which belonged to her mother - because a daughter is supposed to receive her mother's things. Jem will receive Atticus' watch because the son will receive gifts from the father.
"Scout: May I see your watch? "To Atticus, My Beloved Husband." Atticus, Jem says this watch is gonna belong to him some day.
Atticus Finch: That's right.
Atticus Finch: Well, it's customary for the boy to have his father's watch.
Scout: What are you gonna give me?
Atticus Finch: Well, I don't know that I have much else of value that belongs to me... But there's a pearl necklace; there's a ring that belonged to your mother. And I've put them away, and they're to be yours."