One that runs through all the series: just how much have the Greek Gods moved from being Jerkass Gods? It is clear they aren't all smiles and rainbows, but a fan can get a lot of mileage about wondering what their specific deal is. Have they become less jerkass over time, or have they become better at hiding it and pretending to be a lot nicer than they actually are? Do they have Blue and Orange Morality by human standards, so trying to go 'good gods and bad gods' isn't quite tenable? Are the Gods just so complicated they can be both loving and distant in equal, genuine measures? Fans can run with a wide variety of interpretations.
One for Zeus specifically: Does he have an Inferiority Superiority Complex as the source of his behavior? He quarrels with Poseidon over who their mother Rhea likes better, is paranoid Poseidon is constantly plotting against him, demands respect when no one respects him, closes off of Olympus due to wounded pride in a later series, and tries to blame others for his mistakes while downplaying his own. He massive ego and demand people respect and obey him comes across as at least partially trying to cover up his own mistakes and knowledge he is a failure that no one likes, particularly compared to a non-senile Ra and Odin, who don't get mocked or defied nearly as much as he does.
Angst? What Angst?: The preteen to teenage main characters aren't nearly as psychologically affected by always having to fight monsters that are trying to kill them so often as they probably would be in real life. At the very least, there's no sign of PTSD. However, given that they have to do this for years on end, it could be safe to assume they just get desensitized by it, or as demigods are more geared to handle such things.
Crack Pairing: Tons. Rachel/Nico, Thalia/Nico, Percy/Clarisse, Rachel/Luke, Apollo/Hermesnote Which is actually technically possible in the source material, especially given the fact that both are involved in same-sex relationships with mortals and other gods, etc.
Die for Our Ship: Rachel is frequently treated to this by Percy/Annabeth fans. It considerably lessened after the last book, however, mostly because she became the new Oracle.
Draco in Leather Pants: Luke Castellan. While he does have a fairly sympathetic backstory and motives, he is still responsible for starting a war in which dozens of demigods and many more mortals died and would end up bringing The End of the World as We Know It had he fully succeeded. The fans tend to bring up his affection towards Annabeth as a redeeming quality, even though it didn't stop him from trying to kill the girl multiple times and effectively torturing her at one point.
Nico di Angelo gets the most love in this category, going from cute, annoying kid to an emo badass over the course of the series.
Fandom still showers attention on his sister, Bianca di Angelo, too. Never mind that she dies in her introductory book. Many fanfics explore either her life after death or her life in a "What if?" scenario should she have lived.
The twins, Artemis and Apollo, also get a lot of love, despite their very minor appearances. Quite a few fanfics are dedicated to them.
Similar to the above, the Hunters of Artemis get a lot of attention in fanfic, especially notable considering that very few members are named. Artemis's lieutenant, Zoë Nightshade, is especially popular.
Calypso has quite the following, and avoids the Die for Our Ship treatment that Rachel first suffered. Many rejoiced when she returned in the sequel series.
"Funny Aneurysm" Moment: Percy and Nico's first meeting is a pretty funny scene where Percy saves Nico's life and Nico proceeds to talk his ear off about mythology. It's significantly harsher after The Heroes Of Olympus showed a glimpse of Nico's POV in that scene —basically, he saw Percy as a mythical hero come to life, which only made his crush on Percy and Percy's failure to save Bianca that much harder for him to accept.
Genius Bonus: It might or might not be intentional, but the mention of a statue of Susan B. Anthony strangling Frederick Douglass in The Last Olympian might be a roundabout reference to how the two civil rights leaders, formerly steadfast allies, became divided over the issue of the Fifteenth Amendment allowing black males to vote, but not women of any race (and to add insult to injury, it was the first Amendment that explicitly mentioned male and only male citizens).
Growing the Beard: Most agree that The Titan's Curse was where the series really began to show its quality.
Percy catching Annabeth when she nearly falls off Olympus becomes cringe worthy after the events of the third The Heroes of Olympus book, Mark of Athena, where he catches her again while she's being dragged by Arachne's threads into Tartarus and this time around, he chooses to fall with her.
Promise (which seems to be the Arc Words in Luke's life), after reading The Diary of Luke Castellan.
The tense relationship between Nico and Percy becomes downright depressing after The House of Hades, where it's revealed Nico was hiding his true feelings about Percy.
It's easy to miss it, but in his introduction scene, Mr. D quips how his father (Zeus) likes punishing him. This is bad enough but it becomes ten times worse when you read the Trials of Apollo.
For that matter, the fact that he named a hippocampus that he developed a particular attachment to "Rainbow".
It is mentioned that monsters (which Tyson technically is) age differently from mortals and he is, sort of, a child.
Towards the end of The Lightning Thief, Grover speculates that he will be reborn as a flower. Along comes Undertale, a video game which also involves a young, sweet-natured goat-like creature dying and being reborn as a flower.
In the same videogame, what happens to a monster if you kill them? They turn into golden dust- just like in this series.
Chiron's cover as Mr. Brunner is a teacher in a wheelchair. In the Nasuverse, the Heroic Spirit Chiron is summoned into a certain Holy Grail War by Fiore, a mage in a wheelchair.
Hell, when Luke gives him the enchanted sneakers, Percy admits the gesture made him blush almost as much as Annabeth. Who he then jibes for being obvious about her crush on Luke.
Percy and Nico share a bit in The Last Olympian. And then in later books, it turns out that Nico actually has/had feelings for Percy.
Travis and Connor Stoll (from the Hermes cabin). Percy and Tyson, too, but the Stolls are almost never without each other. Of course, the Stolls are also full siblings, which makes it kinda squicky.
Inferred Holocaust: In the first book when Percy, Annabeth, and Grover face Hades, the god of the dead becomes so enraged that he causes not one but two earthquakes, and Percy remarks that it will not have been a peaceful night in L.A. When the kids resurface, Los Angeles is burning. No casualty numbers are given, but it could qualify as a "Funny Aneurysm" Moment, as Hades had just finished complaining about how the Underworld was overcrowded and didn't need more subjects. Kinda ironic.
Manhattan sounds pretty bad by the end of the last book. Everyone had fallen asleep, which probably includes people who were cooking things, people who were in the hospital getting surgery, etc. And Fifth Avenue is a disaster.
I Knew It!: Who is the lightning thief? Could it possibly be Luke, the swordsman who is a son of the god of thieves, and has a scar running across his face?
Launcher of a Thousand Ships: Percy Jackson, of course. Be they mortal, demigod, god, or monster, Percy has been shipped with practically every named character in the series.
Les Yay: Silena and Clarisse, Annabeth and Thalia, Zoë and Bianca. And just... the Hunters as a whole.
Clarisse and Silena in particular considering they play out the relationship between Achilles and Patroclus to its tragic conclusion. And we all know about them.
Magnificent Bastard: Luke Castellan, while introduced as a friendly mentor figure, is The Dragon to Kronos. Holding a resentment towards the gods due to feeling abandoned by his father, Hermes, Luke manipulated countless other demigods into joining Kronos's forces, convincing Silena Beauregard to act as his spy by tricking her into believing nobody at camp would get hurt. Under Kronos's orders, Luke steals both Zeus's Master Bolt and Hades's Helm of Darkness in an attempt to start a war between the gods, convincing Ares to help him when the war god catches him. In The Lightning Thief, when Percy is claimed by Poseidon, Luke summons a hellhound to trick Percy and Chiron into believing that Hades is the true culprit. When Percy succeeds in his quest to find and return Zeus's Master Bolt, Luke lures Percy into the woods and poisons him with a pit scorpion, not taking the risk Percy beating him in a fight. In The Titans Curse, to have her serve as bait for the goddess, Artemis, Luke risks his life by holding the sky to trick Annabeth into taking his place when she holds it to save his life. In The Battle of the Labyrinth, Luke persuades the inventor, Daedalus, into giving him the string of Ariadne so that he can find a route for Kronos's forces to invade Camp Half-Blood. Luke later allows the spirit of Kronos to take over his body after gaining nigh-invulnerability from the River Styx. Despite Luke's manipulative nature, he truly believed he was doing what was best for demigods, and when he realized how much he hurt Annabeth, ends his own life to stop Kronos.
Narm: There's a reason why Antonio Caparo's art for the series is reviled, with special mention often being given to his artwork of Nico.
Rewatch Bonus: For those that read the sequel series, every single scene regarding Nico and Percy takes on a new meaning once you learn Nico had a crush on Percy since the day he was rescued.
Sailor Earth: With the sheer number of Greek Gods, major and minor, with dozens or hundreds of unclaimed children with many of them fighting on the side of Kronos it is very easy to begin imagining any number of demigod characters. This got taken even further with the advent of the gods' Roman counterparts, creating even more possibilities.
The Un-Twist: The fulfilling of the prophecy in The Titan's Curse It says "One will be lost" and everybody assumes it means someone will die. But it never outright states dying, so with the usual Prophecy Twist you could expect Bianca to be lost and later found. It turns out she gets Killed Off for Real.
The mystery of who Percy's dad was could've been resolved immediately if someone had noticed Percy's control of water (at the fountain on the school field trip and in the bathroom when Clarisse was going to dunk his head). But somehow, no one put two and two together. What were their guesses before Poseidon claimed him?
Grover (best friend who saw the fountain, presumably heard about the toilet incident, and sensed how strong Percy's aura is): Nemesis
Annabeth (demigoddess of battle and wisdom, who was there to witness the toilet water incident and was drenched herself): Zeus (who rules the skies)
It was easy to narrow down the suspects in the mystery of the theft of the lightning bolt, considering that the theft took place on the one day of the year that the campers were allowed to visit Olympus (see I Knew It! above). But Chiron thought that the theft was conducted by Hades because...he's big and scary and evil looking.
Another one for Chiron: he knew about the theft for months, but waited until there were only TEN DAYS LEFT to do something about it.
The Woobie: Everyone not named Gabe, Ares, or Kronos:
Luke's mother, May Castellan, also deserves a special mention here.
In The Last Olympian, the narration actually says that Hermes "looked like he needed a hug." If that's not baiting woobiedom, nothing is.
MRS. JACKSON. Ridiculously nice, sweet, and caring, and yet gets get stuck with an abusive slob of a husband and serious financial problems. Luckily, things get much better.
Even Dionysus has a little of this after he revealed his relationship with Ariadne to Percy in the third book, and after one of his sons dies in the fourth book. And then you get to his chapter in Percy Jackson's Greek Gods...
Tropes which only apply to the films:
Angst? What Angst?: Percy seems to handle the sudden death / disappearance of his mother very well...
Anti-Climax Boss: Kronos in Sea of Monsters, possibly intentional because the rest of the books might not be adapted to film.
The Sea of Monsters somehow not being in Poseidon's domain. Ignoring that this is in complete contrast to the books, the Greek myths, and basic logic, two of Percy's other powers (water healing him and his navigational intuition) are later shown to work fine within the Sea of Monsters, making this seem like even more of a lazy excuse for why he couldn't control the water to save himself and his friends from Charybdis.
Percy lamenting Tyson's "death", especially in contrast to the original book. In the book, he knew Tyson from school for some time before they ended up at camp and he found out they were brothers, he had to endure constant ridicule from the other campers over the relationship, Tyson was gone from the story for much longer than he was in the film, and Percy didn't spend nearly as much time reflecting on it. In the movie, the two have only known each other for a day or two, the only character who teases him about it is Clarisse, and Tyson is only gone for a few minutes of screentime, yet his sacrifice appears to drive Percy over the Despair Event Horizon until his friends manage to talk him out of it. As with the above, it comes off as a cheap excuse so that when Tyson does come back, Percy can waste time hugging it out with him instead of preventing the resurrection of Kronos.
Awesome Music: Despite the films shortcomings, they both have great soundtracks.
Badass Decay: Although her actual status as a badass is somewhat disputed (see Broken Base below), Annabeth still does a lot of things in the first movie and is actually helpful in several situations. Come the sequel and she does absolutely squat anymore despite tagging along as a main character for the whole film. Her position as the abrasive Action Girl has been taken over by Clarisse while Annabeth has become little more than a constant Damsel in Distress.
Whether or not Annabeth counts as a Faux Action Girl. Those who claim she is point to the fact that she has to be saved from Medusa by Percy, doesn't figure out the Lotus hallucination until after they're outside it and then does nothing in the climax - despite claiming to have lots of experience with battle strategy at camp. Those who say she isn't point to her beating Percy in their first duel (and he needs a power boost from water to match her), being the one to use Car Fu against Medusa, successfully shooting some guards and the hydra and merely being Overshadowed by Awesome - since only one person can wear Hermes's sneakers. Rule of Cautious Editing Judgment really applies here.
Which of the movies is better or worse and which is the most faithful adaptation.
There's something particularly aggrivating with Hades's Adaptational Villainy. While his morality was ambiguous and he was actually decent in comparison to Zeus in the books, he's given the Everybody Hates Hades treatment from the writers. It certainly doesn't help that the Underworld is designed to look like Hell (literally), thus implying that he's the Devil.
Persephone's strange attraction towards Grover. Most incarnations show her to be Happily Married to Hades, but the film very clearly states that she's unhappy with him.
Evil Is Sexy: Medusa. Of course, she is played by Uma Thurman.
Nightmare Fuel: The scene in Medusa's shop. While it isn't that creepy in the book (there the scene is a lot less ominous up until the Uncle Ferdinand bit), the movie added a panicking woman whose husband had already been turned to stone and she's hysterical with fright. She eventually gets turned into stone too, and the only mention that's made of it is when Annabeth explains how she scraped her wrist (the woman had been holding her wrist when she had been turned to stone and Annabeth had to pull her hand free).