This has been the story of my life, [Jason] thought bitterly. Everyone had always watched him, expecting him to lead the way. (3.18)
Just because someone is courageous doesn't mean they're necessarily happy about it. The problem with Jason is that, before he met all his current friends, he seems to have been surrounded by cowardly people who wanted him to do everything himself.
[Reyna] could only share the qualities she already possessed, and she couldn't help anyone who wasn't worthy of being a hero. (5.38)
Reyna has an interesting superpower: she can enhance someone else's strengths if she has the same ones. Reyna has courage in spades, so she can bring that out in any hero.
[Leo] loved being told that something was impossible. […] He just couldn't resist the challenge. (9.117)
Leo is always determined to succeed, and in this book, he ends up with the ultimate challenge: dying and coming back to life.
"What we want is down there. These pits are the way in. I'll have to jump." (19.66)
Piper does her best Strong Bad impersonation and dives into the unknown. Pretty much every character gets a scene in this book to commit an act of pure blind bravery.
Reyna would make sure Nico was recognized for his bravery. (22.30)
Reyna is majorly brave, so she respects bravery in others. It might be the quality she requires most in a friend.
[Reyna] flung her cloak in front of her and fell on the explosive arrow, determined to shield Blackjack and the other pegasi, and hopefully protect the mortals sleeping belowdecks. She had no idea whether her body would contain the explosion, whether her cloak could smother the flames, but it was her best chance to save her friends and her mission. (39.37)
Here's Reyna's moment of bravery. Leo might have talked about figuratively falling on a metaphorical sword, but Reyna literally leaps onto an actual bomb…to shield a bunch of winged horses. Now that's brave.
According to legend, Athena sometimes cut pieces off her mantle and draped them over statues in her temples, or over her chosen heroes, to shield them. (39.44)
Athena notices Reyna's bravery and rewards her with an item that lets her be even more courageous: an invincible cloak. Although shouldn't cowards be given an invincible cloak so they could be more brave? Hmmm.
Piper followed her gut. She attacked. (43.21)
Even though Piper's strength isn't actual physical power, she still charges headfirst into the giant horde to fight monsters.
I am Nico di Angelo, son of Hades. I control the shadows. They do not control me. (45.46)
It takes courage to stand up to others, but it takes even more courage to stand up to yourself. Nico has to face his past and realize that he can control his own fate.
"You have proven yourself a true hero." (49.48)
Jason gets to hear these words not from just anyone, but from his own father. His father who just happens to be Zeus, king of the gods. Zeus is kind of an authority on heroism, so this really means something to Jason.