How is Percy Jackson a hero in The Lightning Thief?

In more modern terms, a hero is an ordinary individual who does incredible things. In antiquity (Greek and Roman), a hero was an individual who had phenomenal or inhuman qualities and thus did inhuman and amazing things.
In The Lightning Thief, the author uses a mixture of both ideas of hero and heroism. Percy begins as an individual who seems mundane, with no particularly incredible skills, although he is out of place. Most heroes, old and modern, are out of place—this is what propels the individual into a heroic journey that allows them to attain the title of hero.
The setting shifts as Percy is, because of unfortunate circumstances, obliged to go on an "adventure" to be the savior of peace (finding Zeus's lightning bolt). This pushes him into the unreal world—into the incredible world where the mundane disappears. This is shifting towards the idea of a hero in terms of ancient Greek and Roman ideals.
One quality that define a hero in modern times is the notion of experiencing fear but overcoming it with courage. Percy experiences fear in the face of his journey but proves to be courageous (this differs from ancient notions that heroes knew no fear at all). Furthermore, a hero is willing to put themselves on the line for the sake of others—the hero is willing to sacrifice themselves to save others, which is the case for Percy, as he faces enemies and extreme danger in his quest.
Ultimately, a hero is able to achieve a particular goal using their own qualities, whether it's saving the world or standing up to someone. Percy is able to return the lightning bolt to Zeus, thus avoiding war, and to save his mother due to his courage and determination.