Piper's snowboarding jacket was flapping wildly, her dark hair all in her face. Jason thought she must've been freezing, but she looked calm and confident—telling the others it would be okay, encouraging them to keep moving. (2.6)
This is very early in the book—Piper doesn't know she's a demigod, and doesn't know she can persuade people, but she's still trying to take care of everyone. Also note that Piper almost certainly wouldn't see herself as calm and confident. Jason sees her courage more clearly than she would.
"Spirits, fear me!" Hedge bellowed, flexing his short arms. Then he looked around and realized there was only Dylan.
"Curse it, boy!" he snapped at Jason. "Didn't you leave some for me? I like a challenge!"
Leo got to his feet, breathing hard. […] "Yo, Coach Supergoat, whatever you call yourself—I just fell down the Grand Canyon! Stop asking for challenges!" (2.55-57)
Leo could be talking to the author here. The book is just one long series of challenges, after all, and while it makes for an entertaining read it's pretty rough on the characters. You can see where even a brave guy like Leo might find it a little tiresome over the course of five hundred pages.
He could turn around now and tell everyone he'd been joking. Psych! Nyssa could go on the quest instead. He could stay at camp and learn to be part of the Hephaestus cabin, but he wondered how long it would be before he looked like his bunkmates—sad, dejected, convinced of his own bad luck. (11.65)
The alternative to courage and standing up for your friends here seems to be bad luck and dejection. Courage is its own reward in part—which perhaps unintentionally undermines courage. After all, if you're better off being courageous, is it really courage? Wouldn't Leo seem more courageous in going if staying behind would be a cheerful prospect rather than a gloomy one?
Instantly Leo knew that Jason and Piper were in trouble.
Walk away now, the voice had urged.
"Not likely," Leo growled. "Gimme the biggest hammer you got."
He reached into his tool belt and pulled out a three-pound club hammer with a double-faced head the size of a baked potato. Then he jumped off the dragon's back and ran toward the warehouse. (23.34-37)
Heroic rescue superhero moment. Leo is more unsure of himself than most superheroes (he's younger with no costume) so his superheroics seem even more courageous. Superman is courageous so regularly that it stops seeming like courage and starts seeming routine.
"She said I was meant to do something important—something that would make or break that big prophecy about the seven demigods. That's what scares me. I don't know if I'm up to it." (26.107)
Failing is even scarier than Cyclopes or monsters. At least to Leo. We would actually probably be more afraid of the Cyclopes.
"I'm always keeping an eye on you, Leo. But talking to you is, um…different."
"You're scared," Leo said.
"Grommets and gears!" the god yelled. "Of course not!"
"Yeah, you're scared." But Leo's anger seeped away. (29.67-70)
Hephaestus shows some courage here by talking to Leo, who is both an organic life form and his son. He'd be even more courageous if he just admitted he was scared though.
"My point is that love is the most powerful motivator in the world. It spurs mortals to greatness. Their noblest, bravest acts are done for love." (39.56)
Aphrodite doesn't point out that a lot of the most horrible acts are done for love too—Greek mythology is filled with terrible stories of unspeakable things done in the name of love. The Medea story (chapter 28) is an example of someone for whom love led to the courage to betray and kill lots of people.
Tristan McLean wasn't supposed to be seen like this. He was a star. He was confident, stylish, suave—always in control. This was the public image he projected. Piper had seen the image falter before. But this was different. Now it was broken, gone. (45.23)
Tristan is a movie star, so this passage is in some ways pointing out that the kind of courage you see in movies (including the Greek action dramas Tristan makes) is a front. But we're also in the middle of a Greek action drama ourselves. The Lost Hero may recognize that there's something not quite real about courageously battling monsters, but it loves it anyway.
"So…we'll start by boasting will we? Just like old times! Very well, demigod. I am Porphyrion, king of the giants, son of Gaea." (50.53)
Porphyrion enjoys the traditional ritualized declaration of courage. He'd probably enjoy reading The Lost Hero too, where all the characters demonstrate courage at each dangerous encounter on schedule. Though it's maybe hard for Porphyrion to be really brave, since he's so big. When he's at risk at the end from Hera, he just scurries off, without even boasting.
"You're going to make us late for breakfast," Drew said, "which means you get to clean the cabin for inspection."
A week ago, Piper would've either punched Drew in the face, or hidden back under the covers. Now she thought about the Cyclopes in Detroit, Medea in Chicago, Midas turning her to gold in Omaha. Looking at Drew, who used to bother her, Piper laughed. (52.6-7)
We know from the beginning of the book that Piper was brave before she fought Cyclopes or Medea. Her adventures have helped her know she's brave, though, and now Drew doesn't even phase her.