It doesn't seem like much really, just a bow and a quiver full of arrows.
But to Katniss, it's the key to survival. She goes hunting with her bow in District 12, heading out past the electric fence to find food for her family. When she gets into the arena, it helps save her life more than once.
As the saga goes on, her bow becomes as much a part of her as her arms or legs. It comes to represent some very specific things about her: self-reliance, for example. That takes on an added importance in the arena, since she doesn't really become a threat until she scores a bow and arrow for herself.
The bow represents something else about Katniss: her defiance. Once she's a Tribute, it lets her stick it in the Capitol's eye more than once. The big moment comes before the Games, when she shoots an arrow right past her captors and through the mouth of a roast pig. That gets Snow's attention, which leads him to label her a threat which… well, spoiler alert, but the rest of the Hunger Games saga hinges a whole lot on how dangerous she becomes. All from that little bow.
Eat your heart out, Excalibur.