In “The Lightning Thief” Percy Jackson is twelve-years-old. He is launched on “the most dangerous quest of his life. With the help of a satyr and a daughter of Athena, Percy must journey across the United States to catch a thief who has stolen the original weapon of mass destruction – Zeus’ master bolt.” Along the way, he must face a host of mythological enemies determined to stop him. Most of all, he must come to terms with a father he has never known, and an Oracle that has warned him of betrayal by a friend. This story is set in the 21st century. We know this because of the setting and because Percy suffers from ADHD. Attention Deficit Hyperactivity Disorder didn’t really become a known syndrome until the late 20th and early 21stcentury. Even though the author submerses Percy in a struggle with the "gods" and mythology the story is a modern retelling of mythology.
The Lightning Thief, which is the first book in the Percy Jackson and the Olympians series, was written by Rick Riordan and published by Miramax Books in June of 2005. The book centers on twelve-year-old Percy Jackson, a seemingly ordinary boy living in New York City, who has his world turned upside down after a class field trip to the Metropolitan Museum of Art, where one of the chaperones shape shifts into a Greek "Fury" (or deity of revenge) and Percy must fight her off using a pen that has transformed into a sword.
Although the year is not specifically acknowledged in the text of the book itself--most likely to give it "staying power" for later generations of readers--we can contextually assume that it is set in or around 2005, or at the very least, in the 21st century.
Rick Riordan's novel series, Percy Jackson and the Olympians (of which The Lightning Thief is a part), is set in "the modern day." This means that at the time of its writing and publication, the series was meant to reflect the settings, cultural mores and other issues of the day - that day being the early 2000s.
A little research shows that The Lightning Theif, the first novel in this series, was written and published in the early 2000s (published in 2005) and that the book includes the subject of ADHD, in part, because author Riordan's young son was dealing with learning issues at the time.
In this way, the temporal (time) setting of the novel is given additional specificity as the novel is dealing with real-world issues drawn from the author's own life at the time the book was being written.
Percy traveling by Amtrak indicates 1971. "The television blared ESPN." Television technology means at least 1920s, broadcasting ESPN advances the date to 1979. "The ADHD part of me wondered, off-task" The prognosis of ADHD can only be beyond 1987 as the addition the "H" in ADHD is in the DSM-III-R. Daedalus having a laptop with color graphics could be past 1991 when this upgrade became available although the magical qualities of these characters makes this data unreliable. Use of a cell phone by one and all further makes the 1990s likely or even the 2000s as they became increasingly prevalent.
This story takes place in modern time. Around the 2000-2004 period is where it starts.