Seeing all the characters get a relatively decent happy ending: the club shooter is living happily with his wife and daughter, Taro and his friends are more or less regular kids, the Men in Black guys are alive and well as electrical repairmen; hell, even Eiri going to work and griping about his job like a regular guy is kind of endearing. Possibly everything is not as it appears, of course.
Also the scene in episode 12 where Alice reminds Lain of her humanity by letting her feel her heartbeat - doki, doki, doki...
"I promise you I'll always be right here. I'm right next to you, forever." So beautiful. Or really creepy, since a little girl you don't know will stalk you for eternity.
Her father's encouragment (even though we know he was only going along with it as part of the facade) of Lain's computer hobbies and reassurances for her are rather sweet in a family that is mostly cold and distant to one another, are generally reassuring In the end, as he is leaving, he admits that he really did grow to love Lain, even knowing what she is and what she was destined to become.
Lain's father coming back to find her after she resets the world telling her she doesn't have to hide behind her childish teddy bear outfit anymore.
Dad: I'll make us some nice tea next time. Hey, with some madeleines.