In Amazing Fantasy, Martin Li is a teenaged Hero known as the Wraith. He's a Grade Skipper who became a Pro Hero at 16 when most aspiring Heroes are just starting their education, a wealthy businessman, and a philanthropist who runs the highly successful F.E.A.S.T. Centers. He also happens to be a hyperactive Genki Guy and an Attention Whore who won't stop singing his own praises.
The fanfic Ashes of the Past has the Team Rocket trio viewed as this by their boss, Giovanni, who acknowledges that despite their oddity they're very effective. This is also true in reality as well, they're as badass (and odd) as they appear.
Tony Stark, who's frankly rather odd and usually up to something strange, but also one of the most brilliant scientists the world has ever seen.
12 years of Azkaban have left Sirius a little bit unhinged, but he's also a clever man and a very talented duelist.
Doctor Strange seems to cultivate an aura of weirdness, making pop-culture references at the oddest of moments (sometimes to quite poignant effect), dresses like a fairy tale wizard and has a sense of humour that could only be described as odd. He is also by far the most dangerous character in the setting, an exceptionally ruthless Magnificent Bastard with ill-defined abilities as a Seer and/or a time traveler (Harry Dresden opines that it's a little from column A and a little from column B - this turns out to be completely correct. He was a seer to begin with, then got altered by the Time Stone), a master of the Butterfly Effect who conducts the colossal Gambit Pile Up like a symphony and can duel Gravemoss on even footing... while multitasking.
The author himself seems to fit into this category, being a total Motor Mouth who seems to be completely aware of his slightly off-beat view of the world and sees it as an advantage.
In The Circle Xenophilius Lovegood is considered an excellent political tutor despite such personal quirks as sniffing the top of Harry's head and asking Neville if he likes pudding as much as Luna does.
Crimson and Emerald: Camie states that if Hawks wasn't such a good and talented student, Shiketsu would have expelled him for his general irreverence.
A Crown of Stars: Daniel is the God-Emperor of Avalon, an Empire which spans a chunk of The Multiverse. He is the Big Good of his realm, powerful, ancient, and ever-concerned with protect his subjects and help people in need. He is also "quirky" to say the least. It's deliberately invoked, however, since interacting with people in a friendly and lighthearted manner helps him keep in touch with humanity; that, and anytime Shinji and Asuka see him he is usually "off duty", making culture pop jokes or wearing Hawaiian shirts while sunbathing.
Evangelion 303: Mari is nuts. She is an Elvis Presley fangirl, sings and makes "Pew, pew" noises during the battles and she will forget about an ongoing mission to try to beat a strong rival. And still she is good enough to give the base's top pilot a run for her money.
Ranma ½ fanfic Girl Days states that the Kunō Clan was full of Bunny Ears Lawyers, such as a WWII fighter pilot who wore a sailor fuku, or a samurai who fought with a live rabbit strapped to the top of his head... but the Nerima branch aren't crazy awesome but just crazy.
Justice League: The Spider has Spider-Man. He may come across as a light-hearted jokester, but the League swiftly recognizes his skills when he manages to escape Batman, trap the Flash with his webbing and evade them for some time in their own headquarters, to say nothing of his genetics expertise being far above anything the rest of the heroes on the team are capable of.
Lighting Candles: At first, Tadashi views Jack Frost as the most incompetent Guardian, as he's brash, irresponsible, constantly disrespects Tadashi, and makes fun of him whenever possible...but as they get to know each other Tadashi realizes that, while he and Jack are very different people (er, spirits), they are equally capable of and dedicated to their mission of protecting the children of the world.
Mortal Kombat Khronicles portrays master Bo' Rai Cho as this among the White Lotus, during the celebration of Liu Kang being chosen to represent them at the tournament. His drunken, gluttonous, and generally boorish behavior does not really gel with the more reserved demeanor of the other members of the order, but his skill and knowledge in martial arts and as a teacher simply make him too valuable for the others to make a fuss over it.
Old Man Henderson: The eponymous character from a Call of Cthulhu session (arguably being played by another BEL). A half-insane habitual drug user with a stuffed parrot permanently affixed to his shoulder who starts investigating the various eldritch cults because he thinks they stole his lawn gnomes, but by campaign's end he kills Hastur.
In The Parts We Play, while she initially resents Effie’s role in the Games, Katniss soon comes to realise that Effie does a lot more than just show up in odd outfits and read out names from a bowl when she sees just how much Effie does 'behind the scenes', musing that Effie probably has to take on more responsibility than most Capitol guides, considering Haymitch's alcoholism.
In Please Stop Eating The Hell Butterflies, many of members of the Gotei 13 are this. They are all talented and skilled, (from Soifon's stealth skills allowing her to sneak crocodiles into baths undetected to Zaraki's fierce fighting abilities and ability to terrify the enemy) but are batshit insane and taxing to deal with at the best of times. Zaraki in particular is an interesting example, because in canon his personality is basically that of a murder hobo who got one of the top jobs because he is basically a one man army. In said fic he is still a murder hobo and also as motivated by lust as anyone else in story (I.e, everyone), but he's also, amazingly, one of the few captains who actually takes the war with Aizen seriously and actually hampers Gin and Aizen's antics. Yamamoto has probably praised Zaraki for actually doing his job more than half the other captains, even as he laments about getting blood soaked paperwork from Zaraki and telling Zaraki to stop bursting through walls instead of using doors.
Dr. Kaminko from Poké Wars: The Pokémon They Carried has a habit of naming his military hardware after Pokémon. Command lets him keep doing this because he's just that damn good at designing weapons.
ALL of the scientists in the Bleach fanfic Project Tatterdemalion fall under this trope: Urahara, Yoruichi, Tessai, Isshin, Masaki. The only reason they're still employed after all their insane prank wars, bizarre personal quirks, and failure to obey military protocol is that their scientific work is unimpeachable. Shunsui does a much better job of keeping it under wraps but he's pretty eccentric for a high-ranking security officer.
Sunny Breeze is absolutely this in Racer and the Geek, even if he's relatively less quirky than many of the examples here. Switches constantly between the three languages he speaks, knows more about firearms than the rest of the bank security outfit put together, drinks spirits like water, personally owns enough gear to outfit a fire team, is afraid of talking to girls, and is a Shell-Shocked Veteran with an itchy trigger finger. He keeps his job for multiple reasons. First, he is one of the few who can do the job. Second, his boss is his longtime friend. Third, he is a very competent fighter with amazing instincts and a strong sense of honor. Fourth, he hasn't crossed the line so such an extent as to demand something be done about him. In one chapter, they discuss that Sunny is likely one single Wham Episode away from being terminated, should anything happen that even being friends with the boss can't settle. Apparently his idea of what to do if being a guard doesn't work out is to return to being a freelance mercenary.
Oobleck, Qrow and Glynda in Ruby and Nora. Drug addicts and alcoholics respectively, still very good at what they do.
Cynthia in Secret Dreamer is just as eccentric as she is in Awakening, but she shows herself to be a capable commander during an unexpected skirmish with the Risen, deftly managing the recruits and striking more than a few decisive blows of her own.
In Sonic X: Dark Chaos, this trope is the pretty much the only reason Sonic and friends even let Eric anywhere near them. Despite being an Idiot SavantCloud Cuckoolander, he's got the technical skills of Tails with the fighting skills of Sonic - and he's genuinely good (most of the time) under all his insanity. This is also the reason why Sonya deeply loves him despite his craziness.
Spider-X features the brutal example of Max Dillon; he's an abusive jerk who hits women, defies Mystique, and thinks he’s better than most of his ‘teammates’, but the Brotherhood tolerate him because of his raw power.
Between his incompetence at the Digimon card game, lack of experience, inability to make effective plans and shy nature, it's easy to write off Takato from the Tamers Forever Series as weak excuse for a Tamer, but his outward awkwardness masks an astonishing degree of competence.
In the woman behind the queenly mask, a private corporation offered renowned boxer Akihiko a college scholarship on the condition he wear clothes. That's not a figure of speech- he was going around shirtless (and wearing a cape) before.
In Tiberium Wars, GDI Commander Karrde finds Lieutenant Kirce James to be pretentious, self-important, and overly dramatic, which annoys him to no end. At the same time, she's able to whip up squadrons of Firehawks, an Ion Cannon, an entire division of GDI Marines, and a thousand Mammoth Tanks for him on short notice, so he doesn't complain at all when it comes to the results.
In the Naruto fanfic True Potential, Doku Sugita, the director of the Kiri rebellions medical corps, is very passionate about wounds, examining each wound to try to find out how it happened. He also believes that a person's blood type reflects their personality and compatibility with others. However, he's also shown to be an exceptional healer.
The entirety of Stark Industries in The War Is Far from Over Now from R&D (who blow things up several times a day) to Accounting (which has not only a monopoly on drinking games, but also the company record for most alcohol consumed in one sitting). New employyes quickly realize that Tony Stark's company is as unusual and eccentric as he is.
Witches Wizards Shadows and Souls: Dumbledore, the eccentric magic school headmaster with a preference for bright colours and terrifying skill in magic gets along swimmingly with Pegasus, who is the equally eccentric creator of Duel Monsters with a love for cartoons and overly fancy clothes. It gets even more bizarre when it's revealed that Pegasus has been teaching Dumbledore how to play the game, and his deck of choice is Elemental Heroes of all things.
Ichigo in The World In Black And White. He's a bizarrely child-like, candy-obsessed Hollow, who is up to something involving notebooks, while still being every bit as overprotective and capable of kicking ass as his canon self.
Discussed in The Wrong Reflection. Prior to the fic Eleya avoided being cashiered for gross insubordination (cussing out Rear Admiral Tuvok and several dignitaries) because she helped save Earth Spacedock and Qo'noS from the Undine immediately after that. She kept her command and even got a medal, but she also ended up with a black mark on her service record that means she'll probably never make admiral (not that she wants to). The author explains the thought process here.
In Zim the Warlord: Irken Reversion, it's noted by several Irkens that Todog is probably the best Elite in the Empire. However, he's also the laziest and most unambitious, completely refusing to act on his potential, which is why he's never made it to Invader status. Zim believes that he'll probably be one of the most successful Warlords due to his ability to get results, despite his laziness.
Xenophilius Lovegood is considered an excellent political tutor despite such personal quirks as sniffing the top of Harry's head and asking Neville if he likes pudding as much as Luna does.