Dictionary.com defines "flutter" as "to move in quick, irregular motions; vibrate."

Based on this definition, the meaning of "Peeta's eyes flutter open" is that Peeta did not just open his eyes smoothly, but that his eyes opened in a series of irregular movements.

Other examples for the word "flutter" are "the butterfly's wings fluttered open" or "the leaves fluttered in the wind." The word is usually used to show a waving back and forth.

Strong writers must use descriptive language so that the readers can imagine themselves in the story. By including "flutter" in the description, Suzanne Collins not only tells the reader that Peeta opens his eyes, she also shows how Peeta opens his eyes.

Writers are often advised to show, rather than tell, the story, and this sentence is a good example of the purpose behind that advice. The image of Peeta waking up would be more difficult to imagine if Collins has simply wrote, "Peeta woke up." With the way it is written, though, the reader can imagine how he woke up.

When Peeta is awakened, he is startled, meaning that he was in a deep state of unconsciousness. As he regained consciousness, his eyes were working on focusing, trying to make sense of the visual stimulus changing from blank (eyes closed) to busy (eyes open). Fluttering eyelids here are a sign that the conscious mind has not yet taken control of the eyelid motor function; fluttering eyelids are indicated by rapid, almost spasming movement.

"Peeta, Peeta, wake up," I say in a soft, singsong voice. His eyelids flutter open and then he jumps...
(Collins, Catching Fire, Google Books)

People can deliberately flutter their eyelashes, and that action is a stereotyped female motion, meant to appear alluring. In this context, it was likely simply a symptom of Peeta going from unconscious to awake, and has no deeper meaning in the text. When in deep sleep, the eyelids often flutter during REM (rapid-eye movement) sleep, indicating dreaming; REM during a doze is sometimes indicated for the vague shapes seen during sleep paralysis.