Tyson talking about how alone he felt on the streets, and even praying to the gods yielded no results. Then the gods sent him Percy.
The reveal that the bits of metal Tyson had been tinkering with for all of book 2 was a special shield for Percy.
A shield that ended up saving Percy's life in the next book, at that.
In a Fridge moment, the Bookends moment near the end. Percy holds his shield over Thalia in an attempt to protect her from the Manticore; at Mount Othrys, she pays him back by using Aegis to shield him from the force of the exploding car.
When the team encounters Hermes in the film version, he (true to the book) requests that they ask Luke to forgive him for the mistakes he's made, only for Percy to tell him that Luke is too far gone for any of that. Hermes's response?
Hermes:Try. If there's one thing I've learned in three-thousand years, it's that you don't give up on family.
At the end of the book, after they have just won the chariot race, Annabeth kisses Percy on the cheek and the cheering crowd goes even more wild in response. Heck, even the fish could see it.
The Titan's Curse
Percy breaking the cycle of half-blood arrogance, after learning what Hercules did to Zoe: "If I'm going to survive, it won't be because I've got a lion-skin cloak. I'm not Hercules."
And later on, Zoe acknowledging the truth of the statement. "You spoke the truth, Percy Jackson. You are nothing like... like Hercules. I am honored that you carry this sword."
Even more heartwarming when you consider that she gave Riptide to Hercules because she thought he was a true hero, brave and strong and noble, and that she was starting to love him. Percy is all of that and more, and it's clear that Zoe does care for him—almost as a brother—after everything they went through together.
Zoe Nightshade sacrificing herself for Artemis. Even though Artemis was a goddess and couldn't really be killed, Zoe didn't even hesitate before throwing herself between Atlas and her lady. Artemis' clear anguish at her lieutenant's death, coupled with the fact that she chose to give her the highest honor imaginable, preserving her spirit in the stars, adds to the tear-jerking legacy Zoe left behind.
And while we're on Zoe, the fact that she sobs — literally breaks down and sobs — after Bianca's death is not to be shrugged off. This girl is over two thousand years old, has had more than her fair share of loss, and has known Bianca for less than a week. But she is a Hunter, and she trusted Zoe to protect her. And Zoe couldn't, and she mourns her loss as if she'd known her a thousand years.
Zoe's entire relationship with Artemis is this, whether you read it as simple loyalty or something more. She was willing to take the entire weight of the sky to rescue her goddess... and Artemis knew it, and wouldn't let her. She didn't even get a chance to open her mouth before Artemis cut her off.
"No! Do not offer, Zoe! I forbid you."
Apollo assuming a disguise to aid the heroes on their quest, and going against Zeus's decree of no-contact in order to do so, all for the sake of helping them save Artemis. As he puts it, no one messes with his little sister and gets away with it!
The Battle of the Labyrinth
Poseidon telling Percy that he was his favorite son.
Calypso letting Percy go. The fact that she loved him enough to let him go, even though she'd spend thousands of years waiting for a new love which would be ripped away again anyways... This also doubles as a major Tear Jerker.
At the end, Percy telling Nico to join in with his birthday party after he realised that he probably hadn't been to one in his entire life.
Also, the fact that, while Nico blames Percy for Bianca's death, he still protects him by calling up a giant rock to block Kronos' way, even though he could've easily done it a second later and got Percy killed.
The Last Olympian
"Hey, show up with an army of undead warriors to save the day, and suddenly you're everybody's best friend." Nico finally feeling accepted.
"And it was pretty much the best underwater kiss of all time." Certainly heartwarming for the Percy/Annabeth fans, but think about this: Percy has finally earned his reward after four years of absolute torture.
Percy gets a hug from his dad for the first time at the end.
Hades being all stunned at the warm welcome he receives from the other Olympians at the end of book five.
Annabeth taking the knife for Percy, and then that entire scene where Percy tells her about his Achilles spot.
A very subtle CMOH is when Percy and Annabeth, riding through the city, in the middle of a huge battle to save the world, stop to help people who have fallen asleep in dangerous places, like in front of cars, and put out fires. The end of the world is going on, and they still care enough to stop and make sure people are safe. Because they're heroes, and that's what heroes do.
Percy's words to Hestia when he gives her Pandora's Box:
Percy: Hestia, I give this to you as an offering.
Hestia, tilting her head: I am the least of the gods. Why would you trust me with this?
[Hestia] smiled. She tookthe jarin her hands and it began to glow. The hearth fire burned a little brighter.
Hestia: Well done, Percy Jackson. May the gods bless you.
Hestia being there in the first place. She takes the effort to personally visit camp (even if gods can be in more than one place at once), all to improve the atmosphere and create a home away from home (or, Hades, for some even a home, period) to these kids who go through such piles of Minotaur dung in their lives (see the unsorted entry under Tear Jerker) that they celebrate at the end of each summer "yay I didn't die this year either!". Because she is the goddess of the hearth and home and by the Olympians everyone deserves to have that.
At the end of the fifth book, after the gods are finished rewarding everyone, the cyclopes form an isle while standing at attention, and before Percy leaves Tyson says this.
Tyson: All hail, Perseus Jackson, Hero of Olympus... and my big brother!
Hades agreeing, at Artemis's request, to aid her hunters who perished in the Battle of Manhattan, by streamlining their applications for Elysium.
Percy Jackson's Greek Gods
Percy Jackson's Greek Gods is a return to Percy's roots as a First-Person Smartass for the most part, but he also pauses frequently to directly address the reader about the Values Dissonance present in the Greek myths:
When Zeus forces himself on Kallisto by posing as her mistress, Artemis, Percy says that Kallisto kept quiet about it out of fear that it was her fault — then informs the reader that it's never your fault if someone does that to you, and you should tell someone, even adding that though he knows blaming Zeus is not the wisest choice he could make, he doesn't care if the big guy upstairs hears, "I call 'em like I see 'em".
He also makes a point out of Dionysus being the patron god of gender confusion, because it's something he has experience with.
He goes on a brief Author Tract about how the story of Pandora is used by men to justify blaming women for everything that's gone wrong in the world, but Pandora was explicitly set up by the gods to fail, literally designed by them to be unable to resist opening the box, and yet they're apparently blameless.
He also shows a great respect for female demigods in either of the myth-books, speaking highly of Otrera and Cyrene, and giving props to Psyche and her bravery ("Loads of people went to the Underworld with a big sword and an attitude. Hell, I went to the Underworld with a big sword and an attitude. But she had no sword or weapon of any kind, she was already tired from the previous trials, and seven months pregnant.") As a primer to one of the great mythological traditions that is possibly going to be a child's first introduction to the Greek myths, it's really, really nice to see Percy (in reality, Riordan) going out of his way to encourage the reader and try to prevent them from internalizing some of the Greeks' less enlightened ideas.
He also never excuses the wrongdoings of any demigod (for example he says saying Hera caused Heracles' madness that made him kill his family is letting him off easy as he has a well-documented case of Hair-Trigger Temper and he killed a teacher in rage when he was 12) or even god (not even the more questionable deeds of his father such as taking revenge on Minos by having his wife fall in love with the sacrificial bull or raping Demeter in horse-form) in both Percy Jackson's Greek Heroes and Percy Jackson's Greek Gods, and at the end of Arachne's story, he asks, "What's the conclusion to draw? Many say it's to never even think you can be as good as a god and not to even try to outdo them but I think that's wrong. Arachne was that good."
Even with gods he has a rocky relationship with, he at least tries to portray the sympathetic bits while simultaneously being ruthless in calling them out when that is in order. As examples, the two deities at the top of his disliked list:
He readily admits Zeus could be a fun and charming guy when he wanted to be and that he had his share of good plans and ideas plus gives him his due during his fights. Doesn't stop Percy from calling him a rapist and outright unfit to rule ("That would be like making me the god of homework and good grades!")
He is the first to admit that Hera has it rough, being cheated on constantly as the goddess of marriage. Percy is also the first to call her tormenting Zeus's paramours and children more than unjust.
When recounting the escapades of Jason and the Argonauts, he comments on the fight of Polydeuces (he of Castor's twin, from the Dioscuri), he says that even if that particular child might not look like much at first glance, underestimating a kid of Zeus is a grave mistake- and gives special props to "my bro" Jason Grace (and presumably also including Thalia in the compliment).
Snark and opinions aside, Percy mostly sticks to the ancient myths he's recounting. The only time he really goes on about a god's modern-day circumstances is when he talks about Amphritite, a Nereid and wife of Poseidon — and Percy's stepmother. While Amphritite was rather cold when we glimpsed her in The Last Olympian, Percy takes a moment to give her kudos for not taking out her issues with her husband on his innocent children, and says she doesn't give him a hard time for leaving his clothes on the floor and she even bakes him cookies— which is "all you can ask of an immortal stepmother."
On the subject of Amphitrite, the thing that worried her about getting involved with Poseidon was not that she was afraid of being cheated on, but that she didn't want a man to lord over her and wanted to be able to do her own things and be her own person in marriage. When he finds this out, Poseidon happily and gladly agrees to respect her boundaries and the children they will have together and they went on to be one of the most well-adjusted couples.
Simply the fact that if there is one thing the big royally screwed-up family of Olympians agrees on, it's that nobody messes with Hestia with impunity, and if anyone tries, all the gods will be after that person faster than you can say "Uh-oh."
There was one time that both Poseidon and Apollo wanted to marry Hestia. Zeus called her over and said she can chose. She broke down in tears saying she will tend to the hearth and do the job she always did but doesn't want to marry anyone at all. Though they don't really understand her reasons, Apollo and Poseidon rescind their proposal and vow to protect her right not to marry, with Zeus sealing the deal.
To be completely honest, Percy’s relationship with his mother is completely this. There is nothing they don’t share and nothing they won’t to for each other.
For his part, Poseidon speaks in glowing terms of Sally, calling her a queen amongst mortals the likes of whom he hasn't met in a millenium, and Sally says Poseidon, who had a huge heart according to her own admission from earlier, offered her a castle under the sea with everything implied (adding "he thought he could erease all my problems with just one gesture of his hand.")
Percy's relationship with his new stepfather Paul and Sally's relationship with him as well could also be seen as this; especially after years of putting up with Smelly Gabe.
Paul asking for Percy's approval to marry Sally, because he thinks that Percy has a right to know, before he asks.
Especially in The Last Olympian when Paul says that he wishes he had Percy's courage to do something incredibly dangerous that could save them all. Percy even says that he doesn't get compliments like that a lot.
It might not come to mind at first, but Zeus's actions after Thalia's sacrificial death for her friends show that there's more to him than just a stubborn and overly-ambitious ruler, not only by preserving her life in the form of a tree, but also by using her to power the monster-repelling barrier around Camp Half-Blood. In doing so, he's provided protection even for demigods whose mere existence he may not approve of, as well as sparing the other gods the loss he ended up suffering. It's a nice way of showing how deep, deep down, the guy's still not above things like empathy or reasonable sense.
Similar to the above, Argos, the 100-eyed-giant is working at the camp. The same Argos who according to the mythology was tasked by Hera to forever guard her sacred grove, also known as the Garden of the Hesperides. The implication being that Hera, forever described as easy to anger and petty in her (rightful) jealousy, lent the best guardsman she could think of to guard demigods, including those who are the fruit of her husband's cheating.
Dionysus is constantly bitching and complaining about being at the Camp because he hates demigods, but he actually gets to spend time and have meaningful interactions with his twin sons Castor and Pollux. We may not see it on-screen, but Dionysus' utter devastation when Castor dies shows he really does love his children.