There are a few reasons as to why Primrose (or Prim) is important in the novel The Hunger Games.
First, Prim was the initial Everdeen to be called for District 12's tribute. If Prim would not have been called, Katniss would never had needed to volunteer to be a part of the Hunger Games in order to save her sister's life.
Second, the relationship between Katniss and Rue (District 11) would never have happened if Rue had not reminded Katniss of Prim.The relationship between Katniss and Rue proved to Katniss that Careers were not the only ones entitled to alliances.
Lastly, the fact that both Rue and Katniss (given Prim was the reason this relationship existed in the first place) have embraced the mockingjay is important because it, eventually, becomes Katniss' symbol and her unending tie to Rue.
Primrose Everdeen, or as she is normally called, Prim, was the person who was actually called to be the female tribute. She is important to the story because if she was never called, then Katniss wouldn't have volunteered and she never would've been able to fight along with Peeta or talk to him, really. Peeta probably would've died in the Hunger Games like most of the other District 12 tributes.
If Prim had never been called, Katniss would have never volunteered. If she never volunteered and Peeta was still called up as tribute their love would never turn into anything, she wouldn't realize how corrupt the Capitol really is, they wouldn't be the last two standing and try to commit suicide in order to defeat the Capitol, and the uprising wouldn't have started. Prim is vital in the other two books based on her healing abilities.
Primrose Everdeen is important because she is Katniss'es younger sister and if not for her then this novel wouldn't have been exciting if not for Katniss and her flame of love and war.