Although having the grandmother tell the story would still personalize the events and feelings of the story, it would not be as powerful because the daughter as narrator better reflects the themes of the story. The daughter's innocence about the war and its causes shows the ultimate cruelty of the war. The effect on readers is also more powerful when we see things from a child's point of view. Being able to experience the feelings of this young girl is heart-wrenching because we feel her pain. The difficult trek and their life at the refugee camp represents the daughter's journey from childhood to adulthood. It is her rite of passage. She has to become a mother to her younger brother at a young age, forcing her into adulthood before her time.
I think the story would be successful no matter who the narrator is, but the author realized how much more powerful the message would be when told from a child's point of view. Having to grow up in the midst of such horrendous conditions can be felt by the reader through the little girl's narration.