When Hà and her family are preparing to leave their home for good, Mother sews each member a backpack in which to carry little more than the essentials. That Hà's mouse-bitten old dolly makes the cut lets us know right away that this is an item that matters a great deal to our main character, and since it is the only personal item she chooses to bring on their journey, and a comfort object at that, we immediately recognize the doll as symbolizing home and the safety Hà's felt there.
This isn't some shiny new toy, either—as we mentioned above, it's mouse-bitten—but Hà loves the doll all the more for her, as Hà calls them, "scars." Some of the doll's wear and tear comes from being loved by Hà, but some of it comes from the time that Hà lent the doll to her neighbor, who left her outside, which is when she got nibbled on. The doll doesn't just symbolize home and safety, then, it also represents the hardships and uncertainty that loom in traveling away from home—an experience that Hà herself has, too. And in this way, the doll also represents Hà herself.
Along these lines, the doll is part of a particularly poignant moment between Hà and her brother Khoi. Khoi has an incredibly difficult time letting go of the dead little chick he hatched when it is discovered on the ship, and upon seeing him in tremendous pain, Hà uses the dolly to ease his suffering and to help him recover. She writes:
I open Mother's white handkerchief.
Inside lies my mouse-bitten doll
her arms wrapped around/the limp fuzzy body of his chick. (2.7.7)
Hà eases Khoi's pain by offering up her doll, wrapping his beloved chick in its arms and then tossing them into the sea together, and though she misses her doll right away, she knows this was the right thing to do. So the doll is also a symbol of self-sacrifice, and of Hà's love for her family and loyalty to them.
Insofar as the doll is representative of her willingness to make personal sacrifice, it is also indicative of Hà's maturation. Dolls are toys, after all, and toys are associated with childhood—so Hà metaphorically throws her childhood overboard when she opts to help Khoi in his moment of suffering, and when she does, we understand that this journey from Saigon to the United States is forcing Hà to grow up.