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.
In House of Hades, Jason's protective instinct towards Nico after the revelation that he's gay (or bi, it was never specifically stated) and had a crush on Percy.
Jason telling Nico after the reveal: "Nico, I've seen a lot of brave things. But what you just did? That was maybe the bravest." Considering the stuff Jason's seen in his lifetime, that's saying something.
And pretty much everything he says to Nico after the reveal. So completely free of judgement.
"If the others found out, you'd have that many more people to back you up, and to unleash the fury of the gods on anybody who gives you trouble."
"Dude! It's not like you've got a choice. It's just who you are."
Leo asking Calypso to weave a fireproof pouch to put Frank's firewood in so that Frank wouldn't have to worry about it catching fire anymore. It also shows just how far Frank and Leo have come as friends.
The moment everyone gets a chance to register that Percy and Annabeth are out of Tartarus. Jason gives Percy a big bear hug, and Piper hugs Annabeth and bursts into tears.
Nico flat-out tells Jason he doesn't like to be touched (although he does say "please"), yet still initiates affection with his half-sister Hazel, taking her hand or kissing her.