It's not a ball bearing; it's a pearl.
At least, that's what it looks like. Recall that in Catching Fire, they're catching seafood in the water. Peeta finds a pearl in an oyster, and he gives it to Katniss as a gift. As she's worried about him in the Capitol, she's holding on to it as a reminder of him, and as a small token of his affection for her.
Here's a picture of the pearl in Katniss's hand in Catching Fire (taken from the Hunger Games Wiki page about the pearl):
I don't exactly recall what the ball looked like, but it seems to fit your description, and I've seen this theory in other forums (and that Wiki article).
I looked it up, and this theory is backed by the books. Here’s an excerpt from chapter 2 of Mockingjay, as Katniss is on a plane towards District 13:
I feel around for the parachute and slide my fingers inside until they close around the pearl. I sit back on my bed cross-legged and find myself rubbing the smooth iridescent surface of the pearl back and forth against my lips. For some reason, it's soothing. A cool kiss from the giver himself.