What does Atticus tell Scout will be hers when she is older?

Atticus promises Scout that she will receive her mother's pearl necklace once she grows up into a fine young lady. Jem is promised his father's watch. It cannot be ignored that the necklace symbolizes southern culture and tradition—it is not merely a passing mention in the book. The necklace, a very clear symbol of femininity, shows what Atticus, Aunt Alexandra, and the whole community expect Scout to grow up to be like. Scout is a bit of a tomboy and hardly remembers her mother. It is, in fact, Jem who remembers her and misses her the most, and the gifts would likely be more appreciated by both of them if they were reversed (that is, if Jem received the necklace to remember his mother by and Scout received the watch).
The necklace shows that southern society expects Scout to grow up to be a "proper lady" like her mother. Whether Scout actually wants to be one is completely irrelevant. The necklace, therefore, exposes the constricted societal views and expectations toward the girls in the South.

Atticus tells Scout that when he passes away, she will have the pearl necklace that once belonged to her late mother. For his part, Jem will receive Atticus's watch. The necklace is a symbol of traditional Southern womanhood, elegant and refined. It's somewhat ironic, then, that it should pass down to Scout, who's such a tomboy. The suggestion is that however unconventional Scout may be in her childhood, when she grows up she'll have to learn how to be a fine, upstanding Southern lady, the kind of which Aunt Alexandra would heartily approve.
Atticus's gift inadvertently gives us insight into the restricted role of Southern women at that time. Scout is a highly intelligent girl, with the potential to be anything she wants to be. Yet the pearl necklace acts as a reminder that whatever she does with her life, she must always conform to society's ideal of what a Southern lady should be.