"... and I can feel myself slipping away, so I just get out one more sentence. 'Stay with me.' As the tendrils of sleep syrup pull me down, I hear him whisper a word back, but don't quite catch it."
Katniss Everdeen, Catching Fire pg 159
Wow. Reliving this scene has me twisted. The first time I read this my frustration with SC blew it's limit. I just wanted to know what the hell Peeta said to her and I didn't think we would ever find out since how many freaking times up to this point did she do something like that. Cut to Mockingjay and all was (N O T) righted.
If you can, even if you are not rereading with us, take a moment to go back and reread this section from her drinking her tea all the way up to that last line.
I really feel that Peeta began to realize there was something more than friendship in her feelings. The level of security and comfort Katniss displays is paralleled only with Prim. He notices everything about her so there's no way this passes him up. Of course he still has confusion because she appears to have serious feelings for Gale and I'm pretty certain Peeta doesn't see himself as deserving of her (uuuh, now I'm thinking of the six months between and how he must have played that over and over in his head *sobbing* please ignore my emotional outbursts) but there's no way Peeta took this moment for granted. She puts his hand against her cheek and smells it. She's drunk on sleep syrup, no frickin way did Peeta not catch her whiffing him like a kitten.
Katniss *asking* him to not go and then repeating that in the positive form of stay with me is peak vulnerability until the I need you beach moment. She's admitting that she wants him to be with her, if only in a general sense. We never see her ask others to be with her, she constantly attempts to maintain a facade around others and keep interactions to bare minimum. She thinks the danger of the sleep syrup lowering her inhibitions is to Peeta but it simply allows her to cross her own barrier and feel what she tries so hard not to. Katniss gets to experience her own emotions, still distantly, but it's an important stepping stone in her burgeoning awareness of her feelings for Peeta.
This moment also functions as a basis of Everlark growing back together. It's one of the few Real moments between them that could not be touched by the Capitol, where Katniss unguardedly displays her attachment to Peeta. It's so meaningful to him that her repeating the words to Peeta during the escape from the sewers is powerful enough (alongside kissing him until she can't breathe) to bring him back from an episode. That is the breaking point in his recovery and it hinges on the memory of this moment. This is why I suggest rereading it and getting a feel for how utterly precious this moment is from Peeta's perspective.