How does Percy Jackson show he is a hero in The Lightning Thief?

A hero is an archetypal character. Percy embodies a hero throughout the course of The Lightning Thief. Heroes were quite common in Greek mythology, which relates to The Lightning Thief because this book can be viewed as a modern take on a Greek myth. A hero's journey is often split into three stages. During the departure, the hero is called to adventure and often refuses the call. This takes place in The Lightning Thief when Percy ends up at Camp Half-Blood and learns of the prophecy. At first, he does not want to accept the adventure, but he is eventually convinced by Annabeth and Chiron. Once the journey begins, he continues to follow the typical hero's journey as he leaves Camp Half-Blood.
The next stage of the hero's journey is the initiation, starting with the road of trials. We see Percy tackle several "trials" as he encounters problems on the St. Louis Arch, is attacked by Medusa, and endures other similar trials. His ultimate goal is to retrieve the stolen bolt of lightning from Hades and return it to Zeus. His reward for these trials is being able to rescue his mother from the underworld.
The final stage of the hero's journey is the return. Percy's return happens when he leaves the Underworld and brings Zeus the stolen lightning bolt. He is able to complete his journey, and he is finally recognized by his father Poseidon.
In the end, Percy is a hero because his actions prevented the gods from going to war.