The Trope Namer is the Cowardly Lion in The Wonderful Wizard of Oz, and he usually tended to lose the "Cowardly" part of his name whenever Dorothy was in danger or something was required that only a Lion could do. The film version examines this trope in some depth, with the Wizard eventually pointing out to the Lion that he was confusing cowardice with common sense. Sometimes, running away is the most sensible option if fighting only gets you pointlessly killed, as the Lion demonstrates when he flees from the Kalidahs in Baum's original book—however, when he has something to protect after joining Dorothy and the others, he not only stands up to them, but volunteers to sacrifice himself to protect the others from them, until Scarecrow comes up with a plan that makes it unnecessary. The way that he manages to find bravery without the Wizard's help (much like how the Scarecrow typically always comes up with good ideas even without a brain) seems to be a big theme of the book: the heroes had what they wanted all along, but never realized it.
Ron Weasley has shades of this, with his fears of his mother's wrath and spiders constantly gnawing at him. It's his desire to help and protect Harry (and Hermione) that keep pushing him forward.
Kell Tainer, from Wraith Squadron, constantly deals with what he considers to be cowardice. When his commanding officer figures it out, he calls it more a case of "extremely strong performance anxiety", as he's not so much afraid of battle as he is of screwing up, particularly when people are counting on him—understandable, if not necessarily desirable, in military service pilots. Though this officer was unaware of when Kell 1. Literally froze up while controlling an X-wing, unable to work the controls, thinking his ship was instead malfunctioning, and 2. Not really aware that Kell was one lever-pull away from just leaving battle until he thought of his girlfriend, and pulled a Big Damn Heroes. Kell definitely fits this trope in his first book. However, he also grows out of the cowardice almost completely.
The novelization of Revenge of the Sith has a villainous example, compromising between General Grievous's two canonical portrayals, as a Hero Killer (in Star Wars: Clone Wars) or a Dirty Coward (in the film), by making him a very dangerous fighter who nonetheless has a healthy respect for his own skin. He'd rather run than fight, but when cornered, as he eventually is by Obi-Wan, he is a match for nearly any member of the Jedi Order.
In Lee Lightner's Warhammer 40,000Space Wolf novel Wolf's Honour, Mikal feels heavily burdened by the role that fell on him when Berek was gravely wounded. He sees the unconscious leader and asks why him — and is enraged when he realizes that the skald, Morgrim, listened to him. When Morgrim says that he will describe this as a warrior paying respects to his lord before battle, Mikal can not believe him and confesses to his doubts and his conviction that Berek could have led them better; Morgrim assures him that Berek felt the same way and that having never shirked his duty, Mikal has not failed.
Yossarian of Catch-22. He views this as justified due to the fact that there are thousands of people he's never met who are trying to kill him, but as he's a World War II bombardier, it's nothing personal.
Rincewind of Discworld, who has an entire life philosophy based on running away. Nonetheless he has saved the Discworld several times, usually because he simply no longer had anywhere to run to.
"Um... that was very kind of you," said Tiffany, but her treacherous Second Thoughts thought: And what would you have done if it had attacked us? She had a momentary picture of Petulia standing in front of some horrible raging thing, but it wasn't as funny as she'd first thought. Petulia would stand in front of it, shaking with terror, her useless amulets clattering, scared almost out of her mind... but not backing away. She'd thought there might be people facing something horrible here, and she'd come anyway."
Horatio Hornblower, in spades. Every time he sends whatever ship he's on into and out of the fire, he's constantly second-guessing himself and feels like he's lying to his admiring crew because he doesn't give them any hint how honestly frightened he is of being killed or maimed. In one battle, an explosive lands on the deck at his feet. He looks down, grimaces, carefully pinches off the fuse, and disposes of the (now harmless) explosive. He then looks up to see that everyone else on the deck is emerging from cover and staring at him like he's 12 feet tall. From that day forward, his crew worships the ground he walks on, but he still thinks of himself as a coward who just did what had to be done.
Nessus, a member of an entire species of cowards, but at one point, he demonstrates where their instinct to turn away from danger may have really come from:
"All in one motion, the puppeteer had spun on his forelegs and lashed out with his single hind leg. His heads were turned backwards and spread wide, Louis remembered, to triangulate on his target. Nessus had accurately kicked a man's heart out through his splintered spine."
Louis Wu from the same novel. He's a self-proclaimed coward who is bad at fighting (something else that he'll be the first to tell you). That said, he managed to out-think a Pak Protector and fix it so the Protector died ugly, just because he pissed Louis off.
The protagonist of Rowan of Rin is afraid of virtually everything, but has learned to cope with this enough to be functional in everyday life. This leaves him the last one standing in a nonlethal Dwindling Party scenario—everyone else turns back rather than face their personal worst fear, but he's faced his fears so often that it's become second nature to him.
The dragon from "The Tale of Custard the Dragon" by Ogden Nash is a coward that is afraid of everything and wants nothing more than to be in a nice safe cage. His owner, a little girl named Belinda, and her other pets are described in the narrative as being very brave and occasionally tease poor Custard. When a well armed vicious pirate invades their home all of their courage fails them. Custard, on the other hand, is so angry that he briefly forgets he's supposed to be afraid and eats the pirate. Afterwards he goes right back to being cowardly and wishing for a nice safe cage.
Martha Abbott (rhymes with rabbit) in Zilpha Keatley Snyder's The Changeling is this. Deprecated as a coward and crybaby by herself and everyone in her family, she does a number of things that frighten the life out of her. She cries, even pukes, but she does them. Few if any children's books illustrate the difference between courage and fearlessness the way this one does.
While Flashman believes himself to be a dirty coward who has only survived because of his willingness to betray anyone and anything to save his skin, he's managed to do a hell of a lot of heroic things in his career. He's gone into the thick of the action again and again, despite being in the grip of self-soiling terror, because he wants to preserve his reputation. But the point is, he does it. He goes into danger again and again, fighting down his fear the whole way. He's a hero, not because he isn't afraid, but because he overcomes his fear. And he's afraid, not because he's a coward, but because he's smart enough to understand that he could die horribly. Doesn't stop him from being a lecher and an all-round bastard though
In Flashman's Lady he is more afraid of failing his kidnapped wife than anything else.
The foppish bard Dandilion in the Witcher series. He is obnoxious and cowardly, but in The Last Wish he mustered a brave face against execution and in Blood of Elves he held his tongue against torture — torture that would have rendered his hands paralyzed and thus unable to play any musical instruments — rather than betray his friend Geralt.
Jacky Faber is a self-proclaimed coward, and insists that her moments of badassery were just combinations of bad luck, coincidence, and what had to be done, but the fact is that those moments of badassery far outnumber any actual cowardice she may display.
In Herman Wouk's books The Winds of War and War and Remembrance, Leslie Slote views himself as a coward, but his drive not to be one inspires him to commit some brave acts, and he ends up dying a hero.
In "Philbert the Fearful," from Jay Williams' The Practical Princess and Other Liberating Fairy Tales, the title character was a knight who much preferred sitting at home by a cozy fire with a good book. Yet when ordered by his doctor to accompany three other knights on a rescue mission to get some exercise, he managed to get a giant to kill itself by accident, drove off a cockatrice by showing it its reflection and held onto an enchanter who changed himself into a number of dangerous forms. (He also got the girl.)
Butters, a nebbish little polka-loving Jewish medical examiner, goes through the first book in which he appears in utter, pants-soiling terror, but nonetheless manages to stand beside full-fledged combat wizard Harry Dresden, and winds up utterly essential to the good guys' eventual victory, as well as to The Dinosaur Incident. In addition, he's got enough strength of principle to stand by his professional opinion that the corpses of Dresden-burned Red Court vampires aren't human remains, even though it gets him sent to a mental institution for six months. When Harry is temporarily out of commission in Ghost Story, Butters steps up, utilizing Bob the spirit to power a number of Magitek gizmos. In Skin Game, he even takes up Shiro's long-vacant position as a Knight of the Cross.
Morty starts the books as a con man with an atrophied magical ectomancy talent. He has to be badgered, threatened, or out-and-out coerced to help the heroes, but at the same time he uses his power to help plenty of lost ghosts and shades—and to keep the ones who've gone homicidally mad from venting their madness on the living. In Ghost Story, he goes one better and stands up to horrible torture from the ghost of the Corpsetaker, resisting her attempts to break him in order to gain his body and the power of the bound ghosts noted above.
Doc Meadows in Sheep's Clothing starts out first skeptical, then terrified of the vampiric threat to Salvation. However, when his ladylove Sarah is attacked and Wolf calls him a coward for wanting to run away, Doc goes into a full-on berserker rage and empties a rifle into the master vampire.
Victor Krohin from Secret City is an extreme example. He's an absolute coward, and Extreme Doormat to boot. Yet, he's also a Reaper, one of those humans who perform superhuman feats under stress (i.e. mother lifting a car of her child, etc). In-univerce, this phenomenon is explained as rare ability of some Chels to subconciously drain ("reap") magic energy from all nearby artifacts and channel it into tremendous physical strength and speed, impliedly surpassing anything in the Secret City (no mean feat, given that half of City's races are already super strong, fast or both). And Victor is unique in that he can do it more than once in a lifetime. So, whenever he's sufficiently scared and has a magic battery with him, he'll crank up to strength level enough to toss a lagre man forty yards vertically upwards, or rip the heads of three vampires off without them reailzing what's happened. In other Words, he's so badass because he's a coward, not despite it.
Chet Morton and Bess Marvin of The Hardy Boys and Nancy Drew stories, respectively. Each of them is easily the most skittish and afraid of their respective circles of friends, but when its necessary, both have been shown to pull through and have their moments.
In the The Mysterious Benedict Society books, Sticky ran away from his parents, often grows jittery or tongue-tied under pressure, and is afraid he'll never live up to his real name, George Washington. However, he always comes through for his friends, such that Reynie, the main protagonist of the series, is proud to call him one of the bravest people he knows.