Wheatley from Portal 2. He's scared of a lot of things (heights, birds, GLaDOS, etc.) and startles very easily. However, he still sneaks you out right under GLaDOS's nose, uses his flashlight despite believing he'd die if he did, detaches himself from his managment rail despite being afraid of heights and being told that this would also kill him, and has the nerve to talk back to GLaDOS in a situation where Chell can't help him. A lot of this likely stems from him being programmed to have bad ideas, and sacrificing yourself to use a flashlight and sounding off to GLaDOS are brave but very boneheaded moves.
Khalid in Baldur's Gate. He's the easily unnerved Henpecked Husband of Jaheira, who is usually the first to have a morale failure and run in a battle (even though he has some of the highest HP) and can hardly form a sentence without stammering. One of his trademark phrases being "If at first I don't succeed the wife won't let me forget..." , but level him up enough and he makes for a pretty decent tank. And you don't want to have to deal with him while he's Dire Charmed.
If you'd like a demonstration of Khalid not being as big a wimp as he seems, just send your group into a random inn and start indiscriminantly slaughtering everyone. Watch as Khalid's usual stutter completely vanishes and he actually sounds sure of himself for the first time in-game.
Luigi in Super Mario Bros., though this is better exemplified in the spin-off games Luigi's Mansion and its sequel; he's clearly terrified half the time in both games due to his fear of ghosts, but manages to save his brother Mario in the first game and the entire time-space continuum in the second.
Both Koops and Flavio in Paper Mario: The Thousand-Year Door. The former got better after defeating the Koopa-munching Hooktail, and the latter gets one moment of badass when he confronts the undead pirate Cortez and convinces him to lend them his ship. Flavio says it the best when he spells out his plan and the others call him insane:
Flavio: I may be a coward and a cad, but I still lead you! I have a duty to protect you!
Even in fan games, this is one of Luigi's traits. During a Bonus Boss with the Headless Horseman in Rakenzarn Tales, after Kyuu's Taking the Bullet moment, he responds with a lighting attack that outright cripples the boss and makes the fight far easier.
The Maw, a game on the Xbox Live Arcade. The title character is a blob who can eat just about anything and keeps getting bigger until it eats the entire planet, but it has a tendency to flee screaming from bigger enemies that posture in a frightening manner.
Yunobo in The Legend of Zelda: Breath of the Wild. His introductory cutscene shows him freaking out when he mistakes Link, a guy barely a quarter of his size, for a monster. He'll also crouch down and use Daruk's Protection whenever he encounters actual monsters. But he nevertheless braves monster-infested areas and lets himself be shot out of cannons when need be, whether to retrieve medicine for the Goron elder or to help Link board Divine Beast Vah Rudania.
Almaz, from Disgaea 3: Absence of Justice. He's a normal human fighting demons; he has a right to be scared. Still manages pretty well for himself when it counts.
Desco from Disgaea 4: A Promise Unforgotten swears she will do anything to protect her big sis Fuka — while standing resolutely behind her (justified in that despite being an Eldritch Abomination, she is also a little kid). She is one of the most badass characters in the game.
The Grunts. They are tiny, weak, and treated like cannon fodder by tyrants who are larger and stronger and far more numerous than the human supersoldiers. On top of that, they breathe methane instead of oxygen; this means they are constantly fighting in poisonous environments and relying on breathers that can be ripped out of their mouths/nostrils. The methane tanks they are forced to carry on their backs for survival are also extremely prone to explode when shot, thus they are also carrying their own death on their backs. These guys have every reason to be terrified, and yet many of them attempt suicidal attacks of bravery to great effect.
Also, the Marines and Army Troopers of the UNSC. It has been frequently stated that were it not for John-117 and the other Spartans, humanity likely would have been already overrun and destroyed. The Covenant are far more advanced in terms of firepower and raw physical strength than the UNSC, as even a single low-ranking Elite can tear through a whole Marine squad on its own. Human soldiers frequently make comments that make it clear they know how outmatched they are and that in all likelihood, they will die horrible deaths at the end of a plasma rifle. Yet, they do their best to fight effectively against these powerful foes. The only time they are seen to crack up is when facing the monster that is The Flood.
Hahn Mahlay from Phantasy Star IV, a meek young scholar who learns the instant death skills Vol and Savol.
The player character in Penumbra. The "Cowardly" part is represented in-game — if he so much as looks at a monster for too long, he'll panic and give away his position. "Lion" is made more apparent as you play the game and see just how much stuff he goes through while staying sane and ready for action.
Dirk the Daring of Dragon's Lair usually attempts to put on a brave face when battling the disturbing and creepy creatures he has to face when saving Princess Daphne, but it's obvious that he's a freaking nervous wreck the entire time.
Poco from Arc the Lad: a musician who's afraid of his own shadow... and who happens to be one of the strongest fighters of The 'Verse.
In Medieval II and Rome: Total War, generals will often bring up that there is no shame in feeling fear before a battle, but it is letting fear rule you which is cowardly.
"There is no shame in fear. There is only shame in letting fear rule you! Try not to look scared, and you will find bravery in your heart!"
"...any man who faces battle without concern, is a moonstruck fool. To be brave is to go forward anyway, no matter how afeared! That is why I go forward, with so many other brave men."
In Final Fantasy VIII, anytime actual work is proposed, a minor character named Zone immediately crouches by the wall and complains about stomach pains. Later, he goes off and buys emergency train tickets for himself, his leader, and the three main characters, not counting on there being one addition to the party. After a moment, he passes his own ticket to her, then immediately finds a corner to hug his stomach in.
Satoshi Mochida of Corpse Partyhates scary stories. But once he finds himself in one, he actually handles himself pretty well.
Daichi in Devil Survivor 2 is always scared of the next demon attack and complains that the others are much more calm than him, but he'll always volunteer to fight and will drive a truck right onto a killer divine being to save his comrades.
Sparx from The Legend of Spyro fits this trope nicely. He often voices his displeasure with the dangers they must constantly face, but he always sticks with Spyro. When he is a victim of Shoo Out the Clowns at the end of the series, he legitimately fights it, even though he knows he has no chance of surviving.
Garry in Ib. Throughout traveling with Ib in the gallery, he freaks out at things that Ib doesn't even bat an eyelash at (falling over in fear when a painting spits at him, or screaming when a vase falls over, causing Ib to tell him that his screaming was scarier than the vase falling over was), but when push comes to shove, he will do things that need doing, be it fighting off a psycho painting with a knife or walking through a room of poisonous gas. And he will willingly put himself on the line if Ib's in danger, and in some endings even dies for her sake.
Yarne from Fire Emblem Awakening is constantly terrified of going extinct, being the last of his race. Despite this (or perhaps because of it), many of the other characters remark on the true ferocity of his fighting when the chips are down.
The third game in the Phoenix Wright: Ace Attorney trilogy has Ron Delite, who's a bit of a meek crybaby, but still manages to maintain a flamboyant Phantom Thief alter-ego in Mask☆DeMasque, and is revealed to have once saved his future wife from a pair of knife-wielding bank robbers by crying and screaming at them until they left.
Brad "Chickenheart" Vickers in Resident Evil ditches his teammates (Jill, Chris, Wesker, and Barry) after the group gets attacked by mutant dogs, taking the team's helicopter and only means of escape, which strands them in the middle of the mountains thanks to his fear. Brad eventually does come back and attempts to contact Jill/Chris through the radio and he finally shows up during the final boss fight while tossing down a rocket launcher to you so that you can kill the monster in one shot.
Isaac from The Binding of Isaac is a small, naked child capable of beating a gigantic adult, various deformed monsters, and eventually even Satan, all while constantly sobbing and regularly curled up in the Troubled Fetal Position.
The Centaur in Glory of Heracles II. When you meet him, he's being picked on by a bunch of kids, and joins you because he thinks tagging along will help him become brave. One of his battle quotes is even crying out that he's scared.
Umineko: When They Cry: Sakutaro is nervous by nature, to the point of being afraid to go outside unless he's quickly able to retreat into Maria's pocket, but when things get serious, he won't back down from protecting her. Doubles as a literal example of this, too.