There were a few references to original myths throughout the movie, starting with Percy sounding somewhat like Perseus, the name of the demigod son of Poseidon (Percy's father) and a mortal woman. Some of the references in the movie include:
Battling the Hydra: In Nashville Percy and his companions have to fight the Hydra, a creature who grows back 2 heads for every one that is removed. This was originally a feat given to Hercules, who was the son of Zeus and another demigod.
Medusa: When looking for the Pearls needed to visit the Underworld Percy travels to a garden kept by Medusa, the gorgon whose eyes can turn anyone who looks into them into stone. In the original myth the hero Perseus (whom Percy was named after) used his shield to see around corners so he could see Medusa without turning to stone. In The Lightning Thief Percy uses his iPhone's shiny casing to see around corners and confront the Gorgon.
Vegas: While traveling through Vegas, Percy and his friends become trapped in a Vegas casino while the waitresses there continuously feed them lotus flower cakes (or some treat). In the Odyssey, the epic poem by Homer, Odysseus and his men fall into a similar trap on an island where the inhabitants constantly feed them lotus flowers. This makes them complacent and makes them not want to leave. Eventually Odysseus has to drag his men back onto the ship, much like Percy snaps back to reality and takes his friends away from the casino.
Knowing these struggles we can see that the newest demigod Percy is faced with some truly epic challenges, but in each fights he manages to find a way to be victorious, and that no obstacle is too great for Percy and his friends (when you're a demigod with the powers of the ocean god Poseidon, of course).
If you want to learn more there are a few other choices, however this should suffice for now. Hope it helps!