The overnight case that contains Petra's "medicine" is actually full of female hormones, like estrogen. In other words, the medicine that helps Petra in her transition from biologically male to female. Petra desperately looks for the case, and when she finds it, she holds onto it even as it's being swept away by a wave—which takes the tree and the girl who climbed it, too. But Petra gets her case.
She clings to her hormones because they keep her looking female. They're essential to helping Petra feel the most like herself. And in the beginning, before Petra is out as transgender, they also represent her hiding from the other girls.
Petra doesn't want to hide, but it's part of the deal she made with the group who will fund her sexual reassignment surgery. Here's the idea: if she places highly in the pageant and then reveals that she's transgender, her national spotlight will challenge the national ideal of what it means to be female. So, a good cause, but a lot of pressure.
When she's caught by Tiara and almost outed to the group by Shanti, Petra takes back her own power, saying, "You want me to sleep somewhere else, fine. Whatever. But I'm not going to pretend to be somebody I'm not. I've done enough of that." (10.72) Way to go, Petra. Plus, her case doesn't matter as much after she comes out. Which is good news in case another wave comes along, too.