We'd like to think that if Katniss did carry one of our lip balms around in the arena, she would have brought it from The Hob in District 12. (I can see it now: A bubbling caldron of melted lip balm ingredients, right next to the stall with Greasy Sae's soup).
Katniss, a bit of a tomboy with straightforward tastes, would likely have gone for the Bare. It's super emollient and would keep her lips soft and protected from the elements, even while hunting (or kissing, or deciding who to kiss). And it gives off just the slightest whiff of cocoa butter, which, in the arena, might remind her of the luxurious chocolate cake she ate in the Capitol.
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