Abby is the second Black. She lives in "the narrowest house in New York" (5.12), which also used to be the home of the poet Edna St. Vincent Millay. Oskar says that Abby is "beautiful" and has "huge boobs" (5.13) which is a compliment coming from a nine-year-old, we guess. They talk about elephants: how they communicate, their memories, and the fact that they remember the voices of loved ones who have died. Kind of like Oskar.
When Abby tells Oskar that she doesn't know his father, Oskar thinks there's something "unsure" (5.15) about it, and he wonders if she's lying. She's not lying, but strangely, she ends up being the clue to the whole mystery anyway. We should have been tipped off by the subtle title shout-out: Oskar gets "incredibly close" (5.32) to her at one point when talking about the key, and her husband shouts "extremely loudly" (5.18) from another room.
Anyway, it turns out that Abby's husband William knows what the key belongs to: his own dead father's safe deposit box. Oskar returns the key to him, which doesn't bring Oskar any closure about his own father. Still, that has to be good karma, right? William's trying to define his relationship with his father, too, and Oskar helps him with that in a way.
William had a bad relationship with his own father, which lets Oskar appreciate his Dad a little more. William says "I'd trade this key for that [kind of] father" (15.95). But this being the tragic, realistic book it is, neither of them can bring their dads back.