Putting on My Thinking Cap

A character has something they do as an aid to thinking. Crosses into Bunny Ears Lawyer territory if it's something really bizarre.

See also Congruent Memory. Can be a specific character tic.

Anime and Manga

 * L in Death Note claims that if he doesn't sit in the odd way he does, his reasoning ability drops by forty percent.
 * Inspector Lunge from Monster often twitches his fingers as though typing. When asked about it, he likens his brain to a computer - by 'hitting the keys', as he puts it, he can file away new information, then he hits them again to recall it.
 * Naruto: Shikamaru's hand movements and sitting position.
 * Yue in Mahou Sensei Negima seems to have the most going on in her head when she's calmly sipping an odd drink of some kind with a slightly blank look in her eyes. Although in dangerous situations where she has to think on her feet, she doesn't do anything in particular.
 * Saki:
 * While Nodoka is already a pretty good Mahjong player, she becomes absolutely phenomenal when she's hugging her penguin plushie, capable of figuring out plays that would bring about the highest efficiency at once. Almost, she's great when she's playing at home online, so hugging her plushy puts her in the same mindset.
 * Saki finds this in taking off her shoes.

Comic Books

 * In older Disney comics, Gyro Gearloose had a thinking cap he'd put on when he needed an idea for an invention. The cap looked like a roof of a house, complete with a little chimney, and there was a bird's nest with three crows on top of the said chimney. The hat seems to have disappeared from the comics at some point, probably because its role as a source of inspiration was replaced by Little Helper, Gyro's robot buddy.

It was revived in a later comic, where he explained that he had indeed put it into storage, the reason for it was that the ideas that it gave were too random to be of much use. It's also shown that it works just as well on anyone else, and naturally Donald Duck "borrowed" it while Gyro wasn't present to try to solve his usual money problems by coming up with some sort of a revolutionary new idea he could sell and patent. Naturally, this being Donald, there's only one possible outcome...
 * He really should have seen that coming.
 * In other comics, Gyro's thinking aid is hitting himself on the head with a hammer!
 * Nextwave: Dirk Anger has some very odd rituals he engages in when he needs to concentrate, or just be alone. Some of them are intended to be potentially lethal for no easily discernible reason.
 * Leonard Le Genie has a number of gadgets to help him concentrate. They usually involve strong coffee, or holding him upside down to improve the blood flow to his brain.

Film
"Ford: This should give him some zest for a couple minutes."
 * A Few Good Men: Tom Cruise's character -- "Where's my bat? I think better with my bat." And later that scene, "He does think better with his bat."
 * In the movie adaptation of The Hitchhikers Guide to The Galaxy, Zaphod Beeblebrox has a literal thinking cap, that keeps his brain powered by lemons once his second head is removed.

Literature

 * Sherlock Holmes had a number of these, including smoking (in "The Red-Headed League", he explicitly rates the difficulty of a problem by how many pipefuls of tobacco he expects to get through before reaching a solution) and playing the violin.
 * Likewise, Erast Fandorin has his jade rosary beads to help his deductions along. They are so prominent, they even got their own origin story in a short story collection (that was named after them).
 * Violet from A Series of Unfortunate Events will tie her hair up in a ribbon when she is concentrating on something.
 * Winnie the Pooh: Pooh Bear thumps his head while saying "think think think". He also had a Thoughtful Spot, where he went specifically to do this.
 * In the Discworld novel Men At Arms, Detritus the troll gets a literal thinking cap, in the form of a clockwork fan attached to his helmet that lets him cool his brain off (trolls think better in cold weather).
 * In Tom Sawyer, Detective, the lesser-known second sequel to The Adventures of Tom Sawyer, narrator Huckleberry Finn observes that the 17-year-old Tom has a tendency to trace a V-shape on his cheek or chin while he thinks.
 * In The Mad Scientists Club stories, you know Henry is working on a brilliant scheme whenever he tilts his chair backward. He says that sends more blood to the rear of his brain, which is where he keeps his best ideas.
 * In C.S. Forester's Horatio Hornblower novels, when the title character needs to think he starts pacing back and forth.
 * The sentient dinosaurs of Dinotopia can think best when they move their feet, due to their original technique of writing via leaving footprints in the sand. Because of this, reading machines for dinosaurs are treadmills which move the scroll the user wants to read.

Live Action TV

 * In The X-Files, when Mulder needs to think really hard about a problem he watches Plan 9 from Outer Space on the reasoning that the film is so bad it shuts down his logic and allows him to make his standard huge leaps of deduction.
 * A number of the Doctors in Doctor Who - the second played music on the flute or recorder, the tenth puts his glasses on, etc. The Eleventh Doctor seemingly has dozens of these, particularly quick (1-2 step) back and forth pacings, slapping his own forehead, and shutting people up.
 * Blue's Clues had the thinking chair.
 * Sen-chan, DekaGreen in Dekaranger has his "thinking pose" where he stands on his hands. This quirk was retained for his American counterpart Bridge Carson in Power Rangers SPD.
 * Scrubs: Dr. Cox holds his hands behind his head or on top of his head when thinking or frustrated.
 * House: House has several such methods, including bouncing his ball in his office as he stares at his whiteboard. He does these things with such frequency that when he is unavailable other people try them out to force an epiphany. It doesn't work. Cuddy once tried bouncing his ball for inspiration and it hit her foot and rolled off.
 * Stargate Atlantis: Rodney sometimes resorts to fingersnapping and head bobbing.
 * BBC's Sherlock, like his literary counterpart plays the violin and uses nicotine patches instead of smoking, and also stops eating when he's figuring out a case. This can go on for days.
 * Worzel Gummidge didn't have a thinking cap, he had a literal thinking head. Being a living scarecrow, he was able to remove his own head and replace it with a more intelligent one when he needed to concentrate on something.

Mythology

 * Whenever Fionn MacCumhaill sucked his thumb, he got access to great wisdom (though this was the result of him attempting to soothe a burn on his thumb from the Salmon of Wisdom as a child).

Newspaper Comics

 * Calvin and Hobbes: Calvin once built a literal thinking cap out of one of his ubiquitous cardboard boxes, some thread, and a colander.
 * FoxTrot: Jason wore an "external brain" (a cap resembling the surface of a brain) while helping Paige with her homework. However, it's pretty clear that he was just doing it to jerk her chain.

Tabletop Games

 * Dungeons and Dragons has the Headband of Intellect.

Theater

 * William Barfée from the The 25th Annual Putnam County Spelling Bee is noted for spelling out the words on the floor with his foot before saying them aloud. He sings a song about it.
 * Lucky in Waiting for Godot.

Video Games

 * Ace Attorney: All playable characters in the series have their own thinking pose.
 * The Sims 2 has an actual Thinking Cap, a baseball hat with a light bulb. Wearing it will make you learn skills and complete intellectual activities very fast. Side effects may include falling unconscious, losing previously obtained skill points, ending up with the cap welded to your head, and fulfilling one or more Fears. Consult your aspiration meter before use.
 * In the Fallout series, most hats give a bonus to Perception. Perception governs skills like Energy Weapons, Explosives and Lockpicking. So if there's a puzzle involving lasers, bombs and tumblers, you might well need a thinking cap. Or Absinthe.

Web Comics

 * Agatha from Girl Genius hums or plays a tune she invented when she wants to think. Judging by Von Mekkan's reaction, it's possible that every single member of her family did that. Judging by her uncle's reaction, it's an instinctive act linked to being a Mad Scientist. Which makes sense, considering.
 * Hannah from Jayden and Crusader has a thinking cap that her boyfriend stole for her from Uncle Sam!

Western Animation

 * The Noodle Dance from PB&J Otter.
 * Family Guy: Peter Griffin has his Thinking Grenades.
 * South Park: When Towelie wants to think, he gets high (unfortunately, the ideas he comes up when he's high only inevitably get him into trouble).
 * In Fleischer Studios' Betty Boop shorts, Grampy has a literal thinking cap, essentially a mortarboard with a blinking Idea Bulb.
 * In his silent cartoons, Felix the Cat paced back and forth with his hands clasped behind his back.
 * In a related ritual, Scrooge McDuck in DuckTales would retreat to a private room to pace around a stylized statue of "The Thinker"; he's done it so many times that the ground around the statue has recessed into a deep pit.
 * Parodied in Avatar: The Last Airbender, where Sokka is desperately looking for something to replace his missing boomerang as his icon and begins insisting on wearing his detective hat and using his bubble pipe while sleuthing.
 * The Ren and Stimpy Show: "I'd better put on my patented Stuponetron helmet."
 * Cyberchase: Jackie's catchphrase is "Make room! I gotta pace!" She gets annoyed if anyone else steals it. The other characters have their own habits; Inez stands on her head, and Matt plays with his yo-yo. Overlapping with Character Tics in Digit's case is the eyeball on his cap which sometimes flashes red.
 * In The Adventures of Jimmy Neutron, Jimmy has his "Think, think, think!" bit. Then it shows us his brain and a few tidbits from earlier (in case you forgot them) that usually help him solve...whatever.

In a Jimmy Neutron/ Fairly Oddparents crossover, Timmy Turner attempts to do this, only to give up half-way with a tired "Okay, bored now".
 * Bugs Bunny puts on his thinking cap to get an idea to rid the island of Japanese Soldiers in the wartime Merrie Melodie cartoon, 'Bugs Bunny Nips the Nips'.

Real Life

 * Swedish movie and theatrical producer Felix Alvo - http://sv.wikipedia.org/wiki/Felix_Alvo - used to work as an acrobat, which might explain why he liked to stand on his head while pondering something (at least according to some colleagues).