Written for one specific reader
Rick Riordan taught middle-school English and history for fifteen years. When his son Haley, then in second grade, was studying Greek myths and asked for more, Riordan made up a story about a boy who discovers he is a DEMIGOD. It took about three nights to tell and Haley asked him to write it down.
Haley had been diagnosed with dyslexia and ADHD. So Riordan gave PERCY both, and then did the thing that made the books matter to a lot of people: he explained them as advantages. The dyslexia is because a demigod's brain is hard-wired for Ancient Greek, so English letters float about. The ADHD is battlefield reflexes, useless in a classroom and essential everywhere else.
Twelve publishers, then a bidding war
The finished manuscript was rejected repeatedly before an agent sold it, and Riordan kept teaching while the first book came out in 2005. He tested chapters on his classes, which is visible in the writing — the jokes land where a twelve-year-old would want them and the chapter titles are deliberately silly. One is called I Accidentally Vaporize My Pre-Algebra Teacher.
The reframing had a measurable effect. Riordan has said the letters that arrive most often are from children with the same diagnoses saying it was the first time a book made those things sound like something other than a problem. Whether or not the myth-brain explanation is scientific — it is not, and he has never claimed otherwise — a hero whose difficulty at school is evidence he is special is a useful thing to hand a struggling reader.
The gods are relocated, not reinvented
The conceit is that the gods move with the centre of Western power, so OLYMPUS sits above the Empire State Building and the underworld of HADES is under Los Angeles. CAMP Half-Blood is on Long Island. CHIRON the centaur runs it, which makes sense — in the actual myths he trains Achilles, Jason and Heracles, so he already had the job. The cabins are assigned by divine parent, which turns the Olympian family tree into a seating plan.
What Riordan mostly does not do is change the source material. The MINOTAUR, the CYCLOPS, PEGASUS, the SATYR, ZEUS and ATHENA and POSEIDON behave roughly as the myths describe them, and the PROPHECY structure copies the Delphic oracle's habit of being technically accurate and completely unhelpful. Our Greek mythology page has the originals.
What is in the grid
PERCY JACKSON himself, son of POSEIDON and holder of the TRIDENT's favour; ANNABETH, daughter of ATHENA and the strategist of the group; GROVER the SATYR; NICO, son of HADES; and RIPTIDE, the sword that folds back into a ballpoint pen and returns to Percy's pocket whenever he loses it.
RIPTIDE is a decent example of how the series works. Its Greek name is Anaklusmos, it was a gift from Poseidon, it will not harm mortals, and it is disguised as the most ordinary object a schoolchild carries. Ancient object, modern packaging, one small joke about always having a pen on you — that is the whole formula, applied to an entire pantheon.