For this question, I will compare Rick Riordan's The Lightning Thief with Campbell's "Monomyth."
The Lightning Thief follows the basic outline of the hero's journey and only diverges in a few instances from "Monomyth." First, we will discuss what is similar between the two works.
Both The Lightning Thief and "Monomyth" embrace three basic steps in the hero's journey: life in an ordinary world, tests and trials in a supernatural or paranormal world, and a return to the ordinary world once evil has been conquered. In The Lightning Thief, Percy is a wise-cracking twelve-year-old boy attending Yancy Academy, a school for troubled children. He is dyslexic, suffers from ADHD (Attention Deficit Hyperactivity Disorder), and usually performs poorly in academics. Our protagonist's normal world is upended in Chapter One when he battles Mrs. Dodds (one of the Furies). Similar to "Monomyth," Percy initially hesitates to involve himself in the mission before him (Campbell calls this the "refusal of the call").
Chapter Three begins Percy's journey into the supernatural world, one in which he discovers that he is a Half-Blood (part human and part god). At Camp Half Blood, Percy receives the call to action. Campbell describes this the "call to adventure." Percy learns that he is Poseidon's son and that he is the only one who can prevent war between his father and Zeus. Essentially, Percy is tasked with the call to retrieve Zeus's thunder bolt from the recesses of the Underworld, which is run by the god Hades. Percy eventually accepts his mission (which Campbell terms the "acceptance of the call").
Unlike the hero's journey in Monomyth, however, our protagonist accepts the call after he crosses the threshold into the supernatural world, not before.
Also, unlike in "Monomyth," Percy does not discover who his true mentor is until he gets to Camp Half-Blood (signifying his cross-over into the supernatural world). At the camp, he recognizes Mr. Brunner (his Latin teacher) as Chiron, the one who trains demigods to battle evil monsters.
At this time, I want to address Campbell's goddess and temptress construct. In "Monomyth," the hero meets and bonds with a goddess or supernatural being who represents the hero's feminine side. The hero may also later meet with a female temptress who temporarily causes him to lose his way during his mission. Sometimes this temptress can also double as a goddess. Riordan's The Lightning Thief doesn't quite incorporate the temptress and goddess construct in the same way.
In the story, Percy bonds with Annabeth; however, she is neither a temptress nor a goddess. Like Percy, Annabeth is a demigod. Her father is a West Point professor, and her mother is the goddess Athena. In the story, Annabeth travels with Percy to the Underworld, and she is by his side when he fights Ares (the god of war). Annabeth is the epitome of the female warrior; it can be said that she is Percy's counterpart. This is the closest The Lightning Thief comes to reconciling the temptress/ goddess construct in "Monomyth."
You may also decide to discuss Campbell's "atonement with a father" construct. We see this in The Lightning Thief when our hero reconciles with his father, Poseidon. Percy meets with his father and Zeus in Chapter 21. By returning the bolt to Zeus, Percy earns Poseidon's gratitude and esteem. In the throne room, Poseidon openly acknowledges Percy as his own son. Percy can finally rest in the knowledge that his father accepts and cherishes him.