Part 3 of the Archives/Gaiden with that allows you to spend a day with the other main cast. You learn more about them then the action-filled main game allows: Rui and Yayoi's true feelings for Atsuki are revealed; we discover Mika's reasons for wanting to be a reporter; Akira talks about what he wants to do in life and his childhood with Mika, Rui and Ryo; Shinji inadvertently reveals why he's such a jerk to others, and you're given the opportunity to encourage him and see him melt a little. Not to mention Nami's feelings for Atsuki and her support of whatever he does in the future.
Episode 12, where Yayoi smacks Rui out of her Heroic BSOD and gives a speech about how Rui shouldn't kill herself because her friends are always there for her. Particularly noteworthy since Yayoi herself suffered terribly from isolation early on, and Atsuki was there for her.
Any scene where the group tries to get together. While it always ends horribly (because someone is in the hospital or someone close to them died), it's a nice thought. They finally succeed in Archive 1.
This troper thinks that Ryo's one sentence towards Hibiki was heartwarming. "Don't go disappearing on me again." Hibiki's response is a sad smile. It makes sense once you finish the game.