Jason meeting up with Thalia, his sister after they'd been separated for years. Bittersweet, though, in that he also finds out some pretty hard things about their mom.
Percy telling Frank he's proud to be related to him and later, Tyson calling Frank "brother" and Percy going along with it.
Frank entrusting his lifeline stick to Hazel.
Grandmother Zhang telling Frank that she's proud of him.
YMMV, but basically anytime Nico was being a big brother to Hazel.
Annabeth was hit pretty hard by Percy's disappearance, especially so soon after they'd finally gotten together. Piper being so sure that she and Jason were boyfriend and girlfriend probably didn't help. Then, mere pages into Son of Neptune, we find out that Annabeth was the only thing Percy remembered.
Percy giving up the Curse of Achilles, and the chance for a long, peaceful, safe life for the chance to find Annabeth.
Heck, any time Percy mentions Annabeth is undeniably heartwarming!
Percy wanting to protect New Rome partly because he wanted a life for him and Annabeth there.
Hazel recalling the best day of her life.
Percy's reaction to the arrival of the Argo II at the end of Son of Neptune.
Come on. Let me introduce you to my other family.
Annabeth visiting Sally Jackson at least once a week while Percy was missing.
Sammy never forgot about Hazel, after all those years.
Piper, very relieved to see Leo, Hazel, and Frank after an attack on the Argo II and losing them in sea, spontaneously jumps off the ship and paddles towards Leo to give him a kiss on the cheek. It shows that Piper can be concern for the others as well. Could also count as a Ship Tease.
It's a small one, but Frank going to ask for Annebeth's help after getting trapped in the Chinese Finger Trap was pretty sweet seeing how he asked her because he trusted her to help and not tell anyone else.
Especially since ones a Greek and the other is a Roman and they've only known each other a few days
Hazel telling Frank he makes a very handsome elephant and then kissing him.
Percy and Annabeth's little date in Mark of Athena.