Chuckie Finster in Rugrats. This might be why the second movie was enjoyed more by moviegoers (it focuses on him)... well, that and a good chunk of it was a Tear Jerker.
Dave the Barbarian is this in its purest form. Best demonstrated in the episode Red Sweater of Courage, where upon donning the eponymous garb he loses all sense of fear and becomes a Blood Knight, faces most of his Rogues Gallery at once, and gets his butt kicked. Upon losing the sweater, he immediately cowers as normal before immediately coming up with a creative plan that uses one of his many, many girly hobbies to defeat the villains and save the day.
Courage in Courage the Cowardly Dog. He's even called "Leone" (Italian word for "lion") in the Italian dub from the saying "courageous like a lion".
Eduardo from Foster's Home for Imaginary Friends. He's easily frightened, but when someone he loves is in trouble, he shows that his muscular upper body and huge horns are not there for show. He was actually created by a little girl to protect herself from street kids, but when he ended up becoming afraid of them, she ended up becoming the one protecting him. She credits his existence as how she became braver (and eventually, a cop).
Bernard in The Rescuers series. He regards himself as a lowly janitor, but Miss Bianca has complete faith in his courage and skill and she is never disappointed in times of need.
The Galaxy Rangers' informant Geezy. He's frequently protesting about how the "hummings" willingly put themselves in danger, even more frequently protesting that Humanity Is Insane...but if the chips are truly down? He'll put his oversized trunk on the line and get the job done.
Filmation's Ghostbusters featured Eddie Spencer. Eddie was always a lot more timid and uncertain about taking on Prime Evil's minions than his teammates Jake and Tracy, but they knew he'd always be there when they needed him.
Fluttershy from My Little Pony: Friendship Is Magic. In "Dragonshy", she's literally paralyzed with fear at the thought of having to confront a grumpy full-grown dragon and convince him to leave Equestria. However, after the dragon attacks her friends, she forgets to be afraid and gives him such a talking-to he breaks down crying. To clarify the scale; she could easily have stood in the dragon's eye socket. Another time, she stared down and verbally berated a cockatrice, which had already turned her friend to stone, while it was in the process of turning her into stone as well. Good thing it started from the back.
And outside the realms of life-threatening fauna, Fluttershy can summon great reserves of self-confidence whenever her friends are in need while usually being very timid and, well, shy. While normally her reaction to public attention ranges from self-conscious awkwardness to full-blown panic attacks, in "Green Isn't Your Color" she continued to work as a very successful fashion model because she thought that's what her friend wanted, even though it made her extremely uncomfortable.
And, she has major confidence issues when it comes to flying, largely due to being bullied when she was young. If she's ever in a position to forget those issues, such as saving a baby bird falling out of a nest, or saving all of Equestria, she can keep up with or even out-fly RAINBOW DASH.
The vultures of Disney's adaption of The Jungle Book tremble before Shere Khan and flutter to the highest tree, when he threatens Mowgli and Baloo however, they quickly swoop in and take the fearsome tiger head on.
Kim Possible's best friend and partner (and later boyfriend) Ron Stoppable continually has to deal with a host of phobias, largely stemming from his childhood summers spent at Camp Wannaweep. Lampshaded by Kim's cousin Joss:
"Letís face it, Kim, you can do anything. So facing all those dangers and villains, well, itís just like you say. No big. A fella filled with that much fear, always charginí into action with you? Seems to me thatís a true hero."
Early tv star Tom Terrific had the ironically-named Mighty Manfred the Wonder Dog, who always suggested avoiding the adventure of the day.
The Sweeps in The Transformers are sometimes portrayed this way, especially in the episode Starscream's Ghost.
Cyclonus (hunting for Octane in the Decepticon Crypt): "How dare you disgrace your ancestors by cowering like a pocket computer!" (to himself) "He's worse than the Sweeps!"
Justified in that a few episodes ago they went to the Autobot Mausoleum, and got their skidplates kicked by Zombie Optimus Prime. So who knows what would have been lurking in the Decepticon Crypt?
Brain the dog fro Inspector Gadget has been known to cower on more than one occasion, though he acts quite brave at other times.
Sunil in Littlest Pet Shop (2012) is afraid of a large number of things, most of them with no explanation. However, on morethanoneoccasion, he finds himself in a dangerous team situation and the only one in a position to do anything. In these situations, he invariably will find a way to resolve the matter and rescue his friends all by himself.
Fry from Futurama, who is scared of almost everything (when he's not being too stupid to understand that there's something to be afraid of) and tends to hide behind Leela or run away when things get hairy, but when it comes down to it will do anything, up to and including sacrificing his own life more than once, for his friends.
Peso on the The Octonauts is always weary of dangerous and often acts nervous and timid. He never, however, shies away when there's a sick or hurt creature who needs his help.
An episode of The Lion Guard has a cowardly leopard named Badili, who is good-natured and hates to fight. He gets chased out of his own territory by a bigger bullying leopard plenty of times. But once he gets training from the Lion Guard, he comes back confident and ready to take back his territory.