When in the Parthenon Museum, Annabeth begins telling Percy a story about her mother, Athena. The story she tells is the Greek myth about how the capital city Athens got its name. The citizens of the city were arguing over which god to name the city after. The two choices were Athena and Poseidon, Percy's father. Both agreed that they would bestow a gift onto the town, and whichever gift was deemed more useful, that giver would have the town named after them. Poseidon stuck his trident into the ground, and created a spring, which the townspeople could use the water from. Unfortunately, the water turned out to be too salty and therefore useless. Athena presented a single olive tree. The people realized the tree had many uses: Its fruit for eating, its wood for building and burning, the oil from its fruit for cooking and for bathing, and the pits of the olives could be planted, therefore bringing up more olive trees. The people agreed the olive tree was more useful, and therefore named the city "Athens", in honor of Athena.
In Greek Mythology Medusa was turned into a Gorgon because her great beauty lead to her being raped by Poseidon, Percy's father, in the temple of Athena, Annabeth's mother, who took offense and cursed her. Hence Medusa's hatred for them, and her reference to unfinished business.
When Chiron is calling for the campers to gather around to go over the rules for Capture the Flag, he yells for Dylan and Paris to stop "lollygagging". Chiron is played by Pierce Brosnan, whose two younger sons are named Dylan and Paris.
Percy was named, by his mother, after Perseus, Zeus' demigod son, and therefore Percy's cousin, who battled against Medusa, cutting off and collecting her head, which Percy also does. Perseus then gives the head to Athena, Annabeth's mother, who places it on her shield, the Aegis.
The hydra Percy faces in the Parthenon Museum in Nashville was a creature in Greek mythology. It was the offspring of Echidna and Typhon. The Greek hero Heracles (known more commonly by his Roman name, "Hercules") defeated the hydra by cutting off most of the heads, save for the one immortal head, having his nephew cauterize the wounds with a burning torch, and burying the remaining head with a large boulder.