6 from the movie 9 is a great example of this trope.
Belle from the Disney version of Beauty and the Beast is at first presented as this, since she's bored of living in a small town and finds escape in the books she reads. The townspeople find her strange because of this. In fact, one line in her intro song (sung by the barber) goes: "Her head's up on some cloud." We end up finding out that's she's really a strong young lady who doesn't hesitate to stand up to the Beast.
Dory, the blue tang from Finding Nemo, fits this trope to a tee, thanks to her short-term memory loss. Among other things, she uses a deadly jellyfish as a trampoline, frequently misremembers the name of the title character, and believes she can speak the language of whales (this one turns out to be true, though). She even mumbles non-sequiturs in her sleep:
Dory: Careful with that hammer... sea monkeys got my money... yes, I'm a natural blue...
Katie, the yellow baby yak from Horton Hears a Who!, in spite of having only one line: "In my world everyone is a pony that eats rainbows and poops butterflies." The rest of the time she'll inexplicably: make a gonk face like she's choking, sit with her back to Horton (he's a kind of Baloo the Bear "teacher" in this adaptation), and finally float off into space... in spite of being a hoofed mammal with no wings. Fittingly, while the other young animals have parents Katie appears to be completely unique.
Every Ice Age film has one of these.
Ice Age has the Dodos, a species that plans to survive the Ice Age which even they themselves say will last for billions of years, with a stockpile of 3 watermelons. Possibly Fridge Brilliance though, because historically they survived much longer than mammoths and sabre-tooth tigers.
Ice Age 2: The Meltdown has Ellie, a mammoth who thinks she's a possum, despite the obvious size difference between her and her "brothers." She got better.
The Incredibles: KARI...THE BAAAAYBEEEESITTEEEER! *eyetwitch*. Then again, after what she had to put up with in babysitting Jack Jack, most people would be a tad deranged.
The LEGO Movie: Uni-Kitty. What makes this example so mega kickass is that one of the Worlds (The one she originates from no less) is actually named Cloud-Cuckoo Land, thus making her a LITERAL Cloud Cuckoo Lander. The way to get to Cloud Cuckoo Land? Through the Lego clouds!!
Lilo & Stitch: Lilo feeds a fish peanut butter sandwiches because she believes that it controls the weather, and makes voodoo dolls of her so-called "friends" out of spoons.
All of the lemurs in Madagascar seem crazy to some degree, but King Julien XIII is definitely the worst. He uses a skeletal hand as a scepter. In the 2nd movie he goes through four different crowns. He clearly comprehends all that goes on around him (which may be why he's King)...
(as god) Mmm very nice, thank you for that sacrifice...
(as Julien) Please have another sacrifice!
(as god) No No I've had enough for today...
(as Julien) Look I will be very insulted if you don't have another sacrifice!
(as god) I DONT WANT ANOTHER SACRIFICE OK?!
(as Julien) Look at you, you look skinny!
(as god) No, I said I've had enough, now clear off!
Hammy from Over the Hedge. For instance, his first idea when the characters wonder what the eponymous hedge is called is to call it Steve, on account of "it's a pretty name."
Tangled: And then there's Rapunzel. With her using a frying pan as a weapon to her walking bare feet through towns, bars and forests, you've gonna admit she definitely looks the part. Not to mention the fact that she doesn't know much about the outside world because she has been locked in a tower her entire life.
Gomez Addams from either the movies or the TV shows (live and animated) of the Addams Family is always just a couple of degrees off plumb. But his total self confidence and manic optimism usually sees him through.
Alice from the Alice in Wonderland (2010) film starts off the movie as one of these, although this doesn't last that long since she immediately turns into the Unfazed Everyman upon entering Wonderland. While dancing with the man her parentsmother had set her up to marry, she gets distracted by a vision of all the women in trousers and the men wearing dresses and subsequently by imagining what it would be like to fly.
Amélie: Amelie and her male counterpart, Nino. They're complete and total oddballs.
Arizona Dream: While most of the cast qualifies, Paul stands out as living on his own private cloud. He's an aspiring actor who's given to falling into any role at any time. If there's a TV on, he'll quote the lines. Go to the theater? He'll hop up in front of the screen and act along with the movie, undaunted by the boos and rain of popcorn. Bump his head on the ceiling fan? That's the perfect time to launch into the Cowardly Lion's monologue from The Wizard of Oz.
Sam and Joon from the movie Benny & Joon. While there are plenty of funny moments with their weirdness, the film also shows the full extent of what having a personal relationship with someone like Joon would entail. Her breakdown on the bus while trying to run away with Sam is particularly tragic.
Thick Kevin from The Boat That Rocked is this to a tee, epitomized when he comes to a Christmas party dressed as the Easter Bunny. Bob from the same movie is a slightly less flamboyant version, as he's mostly just a weird guy who's off in his own world.
Kitten (Cillian Murphy) from Breakfast on Pluto has definite Cuckoolander tendencies, despite dropping the occasional Deadpan Snark when vexed. She even lives in a "small elfin dwelling on Wimbledon Common" for some of the movie.
Marty, in The Cabin in the Woods, begins the movie as a typical stoner whose statements and actions while high are Cloudcuckoolander-ish but by the end of the movie, he is the only one who sees things clearly.
See, my father is Zozo, the dog at the gates of hell.
Satan says you need more color!
Grandma Georgina in the most recent Charlie and the Chocolate Factory. Justified by the implication that she can't really hear that well, so often has no idea what's going on. Doesn't make her character any less funny, though.
The Christmas Bunny: Mrs. Ross, the Bunny Lady. She's also something a Crazy Cat Lady, if you consider the amount of bunnies she cares for.
In-Universe in Cinderella (2015), Drisella and Anastasia see Ella as this, due to the latter's tendency to talk to mice.
Gracie Allen's ditz persona frequently slipped into this type. Sample dialogue from College Swing:
Hubert: Everything makes me think of Love, Gracie. (she leans gently on his shoulder) What are you thinking about? Gracie:(sighs rapturously) Clams. Hubert: Aren't they beautiful? I hope I don't make you think of clams. Gracie: Oh no, no. I was just thinking, if we were clams, we'd never have to take our shoes off. Wouldn't that be wonderful?
Peter Lorre's character in Crack-Up, Colonel Gimpy, is a well-loved eccentric who becomes the "mascot" of an airfield he wanders around in, and later sneaks aboard the protagonists' airplane insisting he has a meeting with a European monarch. He's also given to quoting Byron and having exchanges like the following:
Col. Gimpy: That really was a magnificent speech. You know, a man says nothing because he's wise, or afraid, or stupid. Which are you? Ace Martin: Wise as an owl. Col. Gimpy: An owl! You know, an owl in daytime, she can't see — like this [blinks rapidly] — but in nighttime, she can. [Pause] Goodbye...
Mark from Empire Records. It's not clear how much is the influence of drugs and how much is just him being special. Who cares, he's Adorkable! Oh, speaking of which, no matter how many drugs you eat, you will never see yourself get eaten by the world maggot during the video for "Saddam A Go-Go."
AJ super glues coins to the floor and not one of his friends gives any indication that this is in anyway unusual for him.
I don't feel that I need to explain my art to you.
Queen Tara comes across this way in her introductory scene in Epic, though it's possibly unintentional. She accuses Ronin of being too serious, but she just comes across as not taking the Boggan threat seriously.
Fred Kwan (Tony Shalhoub's character) in Galaxy Quest, with such lines as "That was a hell of a thing" after traveling through hyperspace for the first time, or, after sucking bad guys out through the airlock, "Sorry, I was - door was a little sticky. Did you see that? I'll get one of my boys up here with a can of WD-40."
Poppy from Happy-Go-Lucky. She is kinda happy her bike was stolen, because she always wanted to get a driving licence. She also stuffs chicken fillets in her bra, because: "I like the way they make me feel. Like a natural woman."
The film adaptations, more than the book-versions, show Bellatrix Lestrange as one of the few malevolent versions of this trope. Fittingly, she's played by Helena Bonham-Carter.
Luna Lovegood is a more benign example—pretty much the canonical Cloudcuckoolander. And J.K. Rowling herself considered Evanna Lynch (already a longtime fan and pen pal of Rowling's) to be one of the best casting decisions in the film.
In the Mood: Largely due to the sheltered life she's lead, the mother of lead character Sonny Wisecarver messes up his life. To stop her teenage son pursuing affairs with older women, she calls on not a relative or a local authority figure, but a radio preacher/huckster who jumps at the chance to demonize Sonny's behavior for all he can milk his audience. When he is in court on some dubious-at-best charges, she misconstrues a question about why these older women would be attracted to her son ; she thinks their 'size' question is about his height and weight - its about his sexual prowess. Her comment that 'he's always been a big boy' causes the judge to order said prowess to be measured - and the results announced in court. By movie's end, it is clear Sonny wouldn't trust his mother to pour milk.
In The Love Guru, the sports commentator played by Stephen Colbert. He is a definite Cloudcuckoolander, whether he's on drugs at the moment or not. On drugs:
Trent: Who do you like in this game, Jay? ...Jay?
Jay: I like the Christmas babies, Trent! And I like the way my skin feels when I wearing my rainbow jacket! [starts making robot noises]
...and then off the drugs:
Jay: I would like to apologize for my behavior earlier.
Trent: I know you're under pressure.
Jay: The kids in school...
Trent: ...you don't have any kids.
Jay: I don't have any kids! And someday I'm gonna die...
The Dag from Mad Max: Fury Road is practically named for this trope. "Dag" is Oceanic slang for someone who is a loveable, kooky oddball.
Jim Carrey's portrayal of The Mask is the Anthropomorphic Personification of the Rule of Funny. Makes sense, since his powers come from Loki, the Norse god of mischief. They manifest as if Tex Avery-style Cartoon Physics worked in Real Life. The Mask takes its chaos from the wearer's own mind. Stanley Ipkiss has a fondness for classic cartoons in general and Tex Avery cartoons in particular, so that's how he expresses The Mask. This is foreshadowed by showing lots of cartoon memorabilia, and a poster of Red Hot Riding Hood on his wall. When less harmless people end up wearing The Mask, the results are... bloody, and generally less slapstick. Always funny, at least for a given value of funny...
In the original Mask comic, the mask actually warped the mind of the wearer to the point where all their actions lead to gruesome, horrible, and sometimes hilarious murders. Stanly Ipkiss kills a lot of people. Remember the Tommy Gun? This time, the thugs are blown to bits.
Karen from Mean Girls. ("My boobs can always tell when it's going to rain. Well, they can always tell when it's raining." And standing in the rain, feeling her breasts: "There's a... 30% chance it's already raining")
Regina counts too. Even if she IS an Alpha Bitch, she's a bit off her rocker.
A lot of the characters in Christopher Guest's mockumentary comedies would fit this trope. To take but one example, Fred Willard's dog-show announcer in Best in Show goes from wondering aloud why one entrant isn't dressed up in a Sherlock Holmes-style deerstalker hat and pipe to asking which dog would make a good wide receiver on a football team. He later asks his on-air partner, apropos of nothing at all, to guess how much he can bench-press.
In fact, Fred Willard's primary role in Guest's ensembles is to be the most obvious Cloudcuckoolander in the cast; which is truly a feat, considering that nearly all of Guest's main characters qualify as Cloudcuckoolanders to varying degrees. According various people who have worked with him, he really likes to play this trope in real life as well. Christopher Guest described him on the Charlie Rose show as a man who "got into character twenty-five years ago, and has never gotten out."
Mrs Brown in Paddington. For starters she tries to convince Judy to go exploring London's sewers with her.
Pain and Gain: Paul's cocaine addiction left him a little bit off.
Predators contains a disturbing take—Noland, a US Air Cavalry soldier from Vietnam, who has been stranded on a hostile alien planet and hunted by an entire race of Blood Knights. He's the only human to have survived several hunting cycles, and he giggles at random, rambles on and on to the new meat, and talks to an imaginary friend. An imaginary friend who tells him to kill the main characters for their equipment. Thankfully, he's too insane to succeed.
Jordan from Real Genius is a hyperactiveinsomniac who in her second scene bursts in on Mitch in the bathroom to give him a sweater she made for him overnight, oblivious to why he might be made nervous by this. She also mentions that her former roommate went crazy and had to transfer schools, although she doesn't know if that had anything to do with her. As she develops into Mitch's Love Interest later in the film, her personality gradually becomes less hyper.
Sara: Wow. This guy's insane. Frank: Well, he thought he was the subject of a secret government mind control project. [beat] As it turns out, he really was being given daily doses of LSD for eleven years. Sara: Well, in that case, he looks great. Frank: Fantastic.
Stacee Jaxx from the movie adaptation of Rock of Ages fits this trope so perfectly he could be considered a textbook example. However, it also coincides with Truth in Television since he's the lead singer of a fictional stereotypical 80's band called Arsenal, and a lot of real rock stars have at some point, gotten so caught up in the Sex, Drugs and Rock & Roll lifestyle that they become disconnected to reality and start to live on some weird sense of logic that no one else understands (not to mention they frequently forget simple things that should be easy to remember). Heavy amounts of drugs and fame will do that to you.
Brickman: The key to being a big league pitcher is the 3 R's: readiness, recuperation, and conditioning!
Brickman: I wrap the cake up in my vomit bag, and Voila!... Breakfast!
Brickman: Some guys ice down their arm after a big game...some say that heat's the way to go...but i have discovered the secret...HOT ICE...HOT ICE...I heat up the ice cubes...IT'S THE BEST OF BOTH WORLDS...
Also Bob Carson, Owner of the Cubs
Ernie: Mr. Carson's last year as Team Owner, he must be really depressed. Bob Carson: Oh Boy, Fish, look a decoder ring... I got it out of the Cracker Jack Box... look it fits on your finger. Larry Fisher: Yeah, Yeah that's great Uncle Bob. Larry Fisher: [Whispers to assistant] That man is turning into a cracker jack.
Rubin Farr from Rubin and Ed (played by Crispin Glover, appropriately enough): Among many other instances, he dances around his cluttered bedroom to classical music while squeaking a cat toy, wears a hubcap on his head like a sunbonnet at one point, and interrupts a business seminar to announce "I am the king of the echo people" (it almost makes sense in context, but only because of a hallucination sequence from earlier in the film). He has the occasional surprising moments of deadpan snark though, and sort of gives Ed a "The Reason You Suck" Speech at one point.
In Scrooged, the Ghost of Christmas Present fits this pretty well, if she's not a perfect fit. She also has Jerk Sue tendencies, minus the feminism.
Nearly all of Stevo's friends and acquaintances in SLC Punk! have some serious personality quirks. Most of the time it's justified, since many of them are shown or hinted to have various degrees of mental issues, troubling family life and pasts, or take hallucinogenic drugs. Or all of those combined.
Aunt Martha of the slasher film Sleepaway Camp is arguably a deconstruction, as her eccentric actions lead to her forcing her nephew, Peter, to pretend to be his sister Angela after Angela and his parents die in a boating accident. Furthermore, whatever methods she used mentally break him to the point he snaps and goes on a homicidal rampage at the camp, eventually surgically modifying himself into a woman and becoming a serial killer.
Will Proudfoot from Son of Rambow is a sweet, innocent and idealistic dreamer whose dazzling and vibrant worlds of adventure are poured like rivers of color onto the pages of an old Bible in pencil and pastel. Sadly, this is also the only healthy emotional outlet he has from losing his father and being raised in an oppressively religious community.
U.S. Bill: This is the basement. Want to see the furnace?
Nightbird: That's okay.
U.S. Bill: It's hot. Don't press your face against it for too long or you get red streaks on you for, like, a month.
Henry Bullock from Splitting Heirs. It's proven by his first scene, where he shows up on roller-skates.
Tank Girl. Played With in the case of the title character. She pretends to be a airhead at times, including making bizarre statements. However, when the chips are down she acts rationally enough - the crazy stuff is just her personality or an act, depending on the situation.
Naydenov in the Russian WW2 film White Tiger thinks that tanks talk to him. Active tanks tell him where shells are coming from, so he can dodge, and wrecked tanks tell him how they met their ends. This is tolerated because he is an excellent tank driver.
Gary from The World's End who thinks that the Bible was a fictional story written by Jesus.