This article contains spoilers through to the end of Mockingjay.
At it's heart, The Hunger Games is the story of a girl fighting to save her sister.
Again and again, Katniss's story comes back to Prim. She volunteers to take her sister's place in the Hunger Games, because she knows Prim cannot survive. She fights to win because she promised Prim that she would. She goes along with the romance with Peeta because Snow has threatened Prim's safety if she doesn't. The series is not just an action/war series with a female protagonist at its center, but one where sisters provide the main drive of the action. Katniss initially does not care about revolution. All she cares about is Prim.
And that dynamic is very compelling, considering that Katniss fails to be as "likeable" as we might expect a female protagonist to be, and Prim is exactly the sort of character we might expect an "action girl" like Katniss to dislike and dismiss. In narrative terms, Katniss and Prim are foils to one another, opposites that highlight each other's strengths and weaknesses. Prim is gentle where Katniss is harsh, scared where Katniss is brave, and brave where Katniss is scared. Both are, at times, the strong, heroic figure, and both are occasionally damsel that needs the other to save them, with actions or with words.
In fact, if Katniss is "not like other girls," then Prim is like "other girls," if "other girls" are taken to be stereotypically feminine. She's small and pretty and sweet. She heals a dying goat back to health and adores an ugly cat that her sister wanted to drown. She even risks her life to ensure that the cat survives when District 13 is bombed. Her greatest character strength is caring for others, whether helping her mother to care for dying people, or caring for animals, or listening to Katniss and offering her advice as she gets older.
And these are true strengths, ones that Katniss does not have. Katniss may be able to kill, something that Prim could not do, but she can't deal with injuries or gore, and she can't always sympathize with others and their motivations. She wants to flee the moment she sees blood. Yet Prim, the seemingly "weaker" sister, faces injury and illness with a stoic, if caring, determination. She isn't scared, she isn't even fazed. She knows what she has to do, and she gets it done. Because, it turns out, "feminine" traits can be rather useful after all, even in war.
Prim provides some softness and humanity to a character who might otherwise seem harsh and selfish and cold. She provides a reason for everything that Katniss does, which may explain why she has to die, from a narrative point of view, in the war's final moments. If Mockingjay wants to show that war is often pointless and cruel, what better way than to have Katniss's reluctant revolution succeed, but have the real reason she fought die just before it does, not because of the actions of the enemy, but because the side she fought for could be just as cruel in the end? To Katniss, Prim symbolizes goodness, even more than Peeta does, so it makes sense that she is torn from her as the story comes to a close. All of her struggle, all her sacrifice, seems to have been for nothing in the end.
Although Katniss is the protagonist of The Hunger Games, Prim is the heart of the story. She is the reason to fight, she is the spark for the revolution, and her calm quiet sweetness provides the contrast that allows us to both appreciate Katniss's strengths and see her weaknesses. If Prim were the damsel to a male Katniss's hero, this might be a problem. Instead, The Hunger Games offers us a defining relationship between two very different, yet very close, female characters. People talk about "Team Peeta" or "Team Gale," but the only relationship that really matters, the relationship that drives the whole story, is that between Katniss and Prim.
And that's more than a little refreshing.