Rue was one of District 11's tributes in the last Hunger Games, and her sad and memorable death during The Hunger Games leaves a very strong impression in both Katniss and Peeta. Rue was just a little kid, like Katniss' sister Prim. She shouldn't have been sent into the arena, and she died there bravely.
Even though she's just a memory, Rue has a strong presence in Catching Fire. We see her come up frequently, for example at the first stop of Katniss and Peeta's Victory Tour, when "Every person in the crowd presses the three middle fingers of their left hand against their lips and extends them to me. It's our sign from District 12, the last good-bye I gave Rue in the arena" (4.87). It's not just Katniss and Peeta who still think of Rue; she is especially remembered in District 11.
Of course, Rue's name means regretful remembrance, and that's what her function is for Katniss and Peeta – particularly when Peeta uses his solo time in front of the Gamemakers to paint a picture of her as she looked in death, covered in flowers. Later Peeta says he doesn't know why he did that: "I'm not sure. I just wanted to hold them accountable, if only for a moment [...] For killing that little girl" (17.18). And isn't that what Katniss wants, too – to hold the Capitol accountable? Refusing to forget Rue is one way to achieve that.