She's not normal, that's for sure. The only thing you can pin down about the Strange Girl is that she's hard to pin down (when she isn't doing the pinning, that is). She's visually distinct from her normal peer group, typically in both physical appearance (she's likely to be an Eerie Pale-Skinned Brunette/Raven Hair, Ivory Skin) and dress, but sometimes only one of the two. As far as demeanor is concerned, she can be anything from Bunny-Ears Lawyer, a Cloudcuckoolander, The Stoic, a world conqueror, The Messiah, The Ophelia, an Emotionless Girl... or quiet, sensible and even friendly. Anything goes as long as it stands out from the series' norm.
The unifying trait of all Strange Girls is that they are protagonists and most importantly, are different from their peers (though rarely will she be completely friendless). Her family or home life will be... interesting to say the least. Usually she is in touch with the supernatural aspects of her setting, even if she doesn't have powers herself. Because of this tendency to experience strange and supernatural adventures, it's not uncommon for their social clique to grow into a veritable Monster Mash of friends they've helped in various adventures, though they'll likely start out the series with at least one or two Unfazed Everymen as completely human friends. Before long she's likely to have seen so much weirdness little if anything will faze her.
This character is very likely to be a Goth or show noticeable Goth influences, both in style of dress and in personality. The Goth subculture is particularly well-suited to this character type because it's made up of people who enjoy strange, morbid things, on top of the fact that being different from the mainstream is one of the things that defines all countercultural subcultures. That last part means that it would also be quite reasonable for a Strange Girl to be a member of the punk or Emo*
The actual subculture, not the caricature of it that named the Emo Teen trope.
Haruhi Suzumiya. "Normal humans don't interest me. If anyone here is an alien, a time traveler, slider, or an esper, then come find me! That is all."
Sasaki might be vying for her position here. She seems far more normal, but is the one who actually does weird Kyon out due to always speaking with male pronouns when talking to boys, and the subject of most of her conversations being bizarre philosophical stuff.
The titular character of Acony, a slightly weird, morbid, purple-haired girl that claims to be 20-something. It doesn't help that ghosts frequently appear around her home.
Urabe Mikoto, the titular character of the manga Mysterious Girlfriend X. Urabe being really, really, really weird is pretty much the entire point of the manga — Urabe is somewhat unemotional and unsocial to the point of being a complete outcast, but seems to have some form of psychic powers — specifically, she knows without a doubt that she will lose her virginity to the main character, Tsubaki Akira (the thought causes her to enter hysterics out of the blue during class), and shares an empathic connection with him that is initiated by spit — a common trick she uses on him when she suspects him of dirty thoughts is to stick her finger in his mouth and suck on it before he can react — she then instantly knows exactly what he was thinking. Similarly, Tsubaki is addicted to her spit, and has to have a dose a day or he suffers withdrawal. She can also cut things apart with superhuman speed and precision with the craft scissors she carries around in the waistband of her panties.
Lavinia of Soukou No Strain draws close to a Cloud Cuckoo Lander in a cast otherwise mostly filled by fairly normal people. Bizarre plots, showing up in Rambo-style battle gear to 'save' Sara, making out with a random girl in the shower (okay, she thought it was Sara, but still) and running around a ship naked after said incident... and this is thought of as a fairly minor extension of her normal personality, attributed to stress.
Natalia aka Belarus from Axis Powers Hetalia. She's a beautiful, small, dainty-looking blonde girl in Elegant Gothic Lolita clothes... and she is very, very frightening. So much that she's the only nation-tan able to scare someone as feared as Russia, who is also considered as this trope (only male).
Romania is stated to be another Rare Male Example. We get some glimpses of it when we have him tell his best friend Bulgaria about his very special former boss with a huge grin.
Tsubaki Kasugano and Yuno Gasai from Mirai Nikki. To start.
Rosario + Vampire: In a Harem Comedy filled with Cute Monster Girls, Mizore the snow-woman stands out. She wears an oversized shirt, striped stockings, and loose belts instead of a school uniform, is stoic, and a not-terribly Covert Pervert who once put blood in her succubus friend's bed to trick her into thinking she'd lost her virginity so she'd have a love-scene with the Chivalrous Pervert who was trying to keep her warm ("but why are you shirtless?!"). And I do mean scene: she hid under their bed with a camera!
Akira Takano from School Rumble. She's frighteningly Genre Savvy and knows and understands everything that's going on to the point of near-omniscience, and regularly has bizarre, goofy moments (like running in out of nowhere to give Yakumo a high five after she accidentally tosses a box of chocolates through Hanai's window, and then disappearing just as quickly,) all without ever breaking her stoic expression. Also, she may or may not be moonlighting as a secret agent.
The series also has a Stange Guy example with Karasuma, who also regularly gets into bizarre antics without ever changing expressions.
Despite her normal clothes and appearance, something about Ayumu Kasuga (also known as Osaka) from Azumanga Daioh seems a little off, though not in a bad way. The rest of the cast - and it's full of colourful individuals - agrees that she is the strangest of them, mostly due to her unique imagination and strange hobbies (like semantics or standing in the rain).
Emily the Strange of Hot Topic fame is actually a cynical anarchist who doesn't like other people and sincerely wants to take over the world one day. She doesn't exactly have any plans for it, and kind of assumes that it'll happen through sheer force of attitude.
Little Gloomy, of the comic of the same name. Kind of. She's the only non-monster in Frightsylvania, which is pretty freaking bizarre. But she's also completely off in a few other ways (she has nails in her house, but no hammer. She needed the nails... but not a hammer.)
Serenity Rose, one of only 50 witches in the world and a borderline MacGuffin Girl in her own series.
Supergirl in Supergirl: Cosmic Adventures in the 8th Grade, who has fantasies about herself as the successful and beloved Moon Supergirl (who stops the Moon Beagle Boys from stealing moon money), insists her nostrils are petite, and treats keeping ice cream cold with her ice breath like a dire responsibility.
Delirium of the Endless of course, from The Sandman. Notable because she manages to be a strange outsider, not only when she travels amongst mortals, but even amongst the other Endless who all are mighty peculiar themselves. The only one who sort of kind of still has a connection to her is Dream, presumably because dreams can get all kinds of weird and non sequitur-ish as well.
Wonder Woman not only fills these criteria in-universe, but is sometimes deemed too strange for many writers to get a firm grasp on.
Most of the girls of Nightmares & Fairytales count, if only because they all seem very fond of a creepy (if well-meaning) doll and get mixed up in horrifying supernatural adventures.
X-23. Created to be a living weapon, Laura doesn't exactly blend in with her peers. She freaks out her cousin Megan on her first day of school by reciting the protocol for bribing and torturing law enforcement officials in French when the teacher asks her if she can speak the language. Later in a biology class she corrects the teacher on the amount of blood in the human body before beginning a lecture on poisons when he asks if she has anything else to share. It really says something when one of the easiest people for her to relate to is a vampire.
In The Hearth Series, nearly every girl is strange. Heidi secretly takes pictures for Elizabeta and, by Word Of God is "not as innocent as she looks". Elizabeta has a major yaoi fetish and a violent temper and handily dispatches a crook on the fly with a Frying Pan of Doom. Both Natalia and Katsuya suffer from different forms of insanity, with Natalia relentlessly stalking her brother and Katsuya going incredibly violent and possessive if she forgets to take her medication.
The residents of Wonderland consider Alice a very strange little girl, while ironically they are far more strange than her.
However, Alice is very strange by real world standards. She pretends to be multiple people, talks to herself, spent several minutes pondering whether cats eat bats or bats eat cats, gives herself advice and ignores it, plays chess with her cats, tries to get her kitten Dinah to pose like a chess piece, and once boxed her own ears after cheating in a game against herself.
Angela from the Inheritance Cycle. She wants to prove that toads don't exist, knows every poison and potion known to man, and always knows what the next big event is, where it is, and somehow always manages to get there on time.
"My name is Mary Katherine Blackwood. I dislike washing myself, and dogs, and noise. I like my sister Constance, and Richard Plantagenet, and Amanita phalloides, the death-cap mushroom. Everyone else in my family is dead."
Tithe is about one whose strangeness is explained; it turns out she's actually a changling pixie, and fairies tend to generate a weirdness aura. She's not entirely sure whether this is awesome or annoying.
Stargirl, by Jerry Spinelli, has the titular Stargirl. Her real name is Susan Caraway, but she decided that Stargirl suits her better. She'll go to school in a kimono, knows everyone's birthday and takes a ukulele to sing Happy Birthday to them in the cafeteria, meditates to be at one with the universe, attends funerals where no one knows her, and follows random people around to watch their lives. No one knows what to make of her.
Olivia Kidney, who talks to dead people. The cute pink cover on Ellen Potter's book is misleading; the book is reminiscent of Coraline and similar creepfests.
The titular character of Daisy Miller is Europe's local freaky, bizarre American girl.
Isabella "Bella" Swan in Twilight. She is The Stoic, the Straight Man, and The Quiet One all at once, hangs around sparkly vampires and American Indian werewolves, and has both magical blood that attracts magical creatures toward her and a magical mental shield power and a vampire boyfriend that keeps her safe from harm.
Rachael Elizabeth Dare from Percy Jackson and the Olympians. One of the main reasons for her strangeness is she is a "clear-sighted" mortal, meaning she can see through the Mist, seeing the true world - monsters, greek gods and all - clearer than even demigods. Also, in The Last Olympian, we learn she is fated to be the next Oracle.
Maggie in How NOT to Be Popular by Jennifer Ziegler is like this, only not supernatural. Her parents are hippies who move anywhere they want to and don't really believe in material possessions. She has been to 10 high schools, and when she gets to her 10th she decides she's had enough and pretends to be weird so that nobody will like her and she won't have to worry about leaving behind friends when she moves.
Invoked, only to be subverted; Maggie actively tries her best to act wierd, but she's actually a very normal girl, even considering her antics things only a 'loser' would do. The most out-of-the-ordinary things about her are that she's a vegetarian and has only ever watched television. Her parents, on the other hand...
Lydia of Caught In The Act by Peter Moore.
Raven of Vampire Kisses is regarded this way by her peers. Her family and best friend are completely used to her antics, though.
The few times Anita Blake (Vampire Hunter) mentions her childhood, it's usually about how her necromancer powers made her a Weirdness Magnet, and creeped people out. In addition to the zombie-raising thing, her personality is quirky, a little geeky, extremely ironic and she frequently voices the punchline to jokes you'd have to be in her head to understand (which we are).
Galaxy of Fear has Tash Arranda, though over the course of the series it tones down a little. She's Force-Sensitive and has always been seen as a bit of a Creepy Child for it.
Eva's mother Consuelo from Eva Luna was regarded as such as a little girl. She grows up as The Stoic, and when she conceives and gives birth to Eva she switches to Sugar And Ice Girl.
Degrassi has Imogen, who is a little odd. At first she didn't match this trope as her antics were malicious and dangerous, manipulating people to get what she wanted. Midway through Season 11 her tone shifted and she just became an odd and excitable goofball, and landing her squarely here as she still had just as many quirky and eccentric moments but without any direct result in mind.
Quinn from Zoey 101 used to be this in the first few episodes, before her character was written to be normal.
"Little Miss Strange"" by the Jimi Hendrix Experience.
"Little miss Strange came out of the darkness Walked across my head, I stood beneath the light I'm talkin' 'bout the dream I had the other night"
The girl being sung of in Lonestar's "Unusually Unusual".
"She had a tattoo above her ankle of a Trident submarine She says it symbolizes awesome power hidden deep within our dreams And I knew the shoes she wasn't wearing, they fit her just right And she said, 'Hey man, did you know, we used to be brothers in some past life?'"
Cameo (the "Word Up" guys) have a song called "She's Strange".
In room 123, she's elusive you see Like the invisible man in drag And when you come to meet her, you'll never greet her She'll be waving her skirt as a flag!
Skye Sweetnam's song "Sharada" is about this kind of girl.
Once upon a time there was a girl You wouldn't really call her typical Had her own definition of cool She lived in her own world She had her own style, her own rules She played along like it was usual Nobody really even knew her name To her life was one big game.
Nemi, a goth environmentalist who gets severe brain-damage around hot guys and has childlike glee when it actually snows enough to go sledding.
The female Malkavian of Vampire The Masquerade: Bloodlines is by far one of the most bizarre of any Strange Girl characters. Since her lines are unvoiced, the player is left to fill in the sound from their own imagination, yet she simultaneously gives the impression of complete silence. Not only is the content of her lines head-spinningly insane, these impressions of muteness and derangement combine to make it seem as if the voices in her head are communicating with everyone else, an impression supported by other characters' reactions to her.
Persona 3 - It's unclear what, exactly, Elizabeth is (the closest we get to an answer is "Resident"), but she doesn't seem to operate by most human rules. Probably because she doesn't know what they are.
Her older sister Margaret from Persona 4 is similarly strange, though perhaps not quite so much so.
Lute from Fire Emblem The Sacred Stones. A very skilled Black Magician Girl who destroys monsters "with love", locks herself in the local library, meets those who go recruit her by checking on them to see if they're monsters, etc. Lampshaded if she gets married to Cultured Badass Kyle, with their common ending saying that Kyle's friends were VERY weirded out by his choice in brides.
Also Sara from Thracia 776, a cute White Magician Girl who asks Leaf if she can join him by saying that she can hear his voice in her head and speaks very nonchalantly about how she hates her grandfather Manfloy for killing her dad before she was even born. Even her caretakers in the Lopto Sect regard her as being weird!
"Did she say something to you? Please forget it. She mumbles strange things from time to time. She can be an odd girl at times."
Bella of The Colour Tuesday looks and sounds like someone out of the 18th century, addresses Kyle as if he were a Knight Errant, and treats Alex like a dirty commoner. The setting is a modern, if dystopian one.
Mirania is described as being a deeply spiritual person... by those who aren't straight-up weirded out by her.
Street Fighter Alpha 3 gives us Ingrid, a girl from the far future who talks like an old woman, weirds out someone as calm and collected as Rose (who actually refers to her as strange), and claims that Bison stole her powers to create the Psycho Drive.
Ghost Trick gives both Lynn and Memry. Lynn is a young detective who takes dying multiple times in stride, going so far as to berate Sissel for not showing up to resurrect her quickly enough for her liking. She also has a tendency to die in pretty odd ways, including inadvertently setting off a deathly Rube Goldberg device, and being crushed by a giant plastic model of a roasted chicken. Memry, meanwhile, is a waitress (actually an undercover cop) who seems to mistakenly be under the belief there's something going on between here and the bartender, has a fondness for bugging the meals of suspicious patrons, and is even called an "odd girl" by several characters (including Lynn)
Ib: Mary, in stark contrast to the titular girl/protagonist - right down to her visibly more vibrant colour scheme. Understandable, given she's a painting, the only semblance of human social having been learnt from books.
Miku in A Profile, contrasting the very normal Miou and the slightly clingy Rizu. While those two are basically normal, Miku has a way of flipping between extremely shy and very cold and dismissive. While this is typical Sugar and Ice Personality behavior, it really stands out in this context.
Kotomi Ichinose from CLANNAD. Extremely smart but has virtually no social skills, to the point where she doesn't even acknowledge people talking to her.
Toko and Stella in Kara No Shoujo. The former has a kind of twisted way of thinking and is sort of bewitching which stands in great contrast to the mostly zombie like student body at her school. The latter is a spacey foreigner both in speech and in the way she doesn't seem entirely grounded.
Ririko in Sharin No Kuni is weird. I... she just... Ahem. Her 'interactions' with Touka when she was a child, her apparent liking of 'teasing' Kenichi, difficulty holding a train of though together and other random bizarre attributes are just the start of it.
To underscore how weird (and Bad Ass) she is, for years she undergoes a punishment that is intended to essentially drive people insane and into suicide. You'd think, at the very least, her personality would be severely twisted by what she went through, like some of the other heroines, right? Heh, nope. Ririko goes back to acting pretty much the same way she always has after one day more or less deciding she's not gonna put up with it anymore.
Maria Ushiromiya from Umineko No Naku Koro Ni is seemingly on speaking terms with many witches, ghosts, talking animals and other strange beasts, but never seems to have mastered interacting with her fellow human beings. As might be expected of someone who was mentored by her, Ange follows suit.
Maya, from the Ace Attorney series. She is a seventeen-year-old girl who is more than a little fond of a show marketed for ten-year-old boys, occasionally channels (and shapeshifts into) her deceased older sister, wears her medium robes as streetwear and thinks nothing of it, tries to get some firemen to spray her down with the hose to help with her medium training, and argues with Phoenix over whether it's proper to call a ladder a ladder or a stepladder.
Marie from T Campbell's Quiltbag, despite not having any super natural abilities (as the setting is more slice-of-life). She makes strange word-salad entries in her journal, has a vast collection of empty bottles strewn about her dorm room, and it is frequently referenced that she is "bad at talking." She even has the look.
Tegan Cassidy in Holiday Wars is a very strange girl and doesn't fit in with anyone from her high school, as seen in this strip
Antimony Carver of Gunnerkrigg Court is a far more "normal" Strange Girl. She's very stoic and has deep-seated issues, but is caring, friendly and ready to lend a hand whether you're a robot, fairy, or ghost. She's visually set apart from her peers by the unusual lipstick and eye-shadow she wears.
Zimmy might also qualify. Her strange skin tone and eyes covered in black gunk make her stand out visually, she wears her school uniform all the time, she's afflicted with the unfortunate combination of some kind of Reality Warper powers and horrifying hallucinations, she doesn't sleep, she's got a Psychic Link with another girl who serves as her Power Nullifier... Honestly, the stuff that can be quantified is probably just the tip of the iceberg. There's a lot of mysteries and open questions surrounding Zimmy as well.
Not to mention what an encounter with her may or may not have done to Jack...
Hilariously enough, it seems to have mostly (after sorting out spider related weirdness) made him fall for her, a neat little send up of the trope, considering how Antimony herself, and even Jack, talk about it. (ie. Her strangeness, and jerkiness, is not particularly romanticized by either.)
It's almost Catch Phrase for her and her best friend Naal'suul to have the exchange "you're weird", and "I know."
Now even Survival of the Fittest has one, of all places! Version 4's Anna Chase is... a little off. To get an idea, she is an Eerie Pale-Skinned Brunette with some level of Perky Goth to her, that is obsessed with horror movies, and is a massiveCloud Cuckoolander. At one point (in her introductory thread in pre-game) she tells her best friend not to eat her brains (It Makes Sense in Context... sort of). And that's not even going into some of her other personality traits. Overall, she is generally seen as somewhat eccentric by other characters.
And then there's Lily Ainsworth, also from v4. She's even more of a Cloud Cuckoolander. Her father was a historian, which caused her to grow up in a British castle most of her life due to his work, and she herself has a tendency to ramble off certain facts as an attempt at conversation, while generally having No Social Skills. Add in her odd dress sense and her perky nature, and most characters also see her as a little odd.
Rebecca Stone from Demo Reel. She's a Nice Girl, but accidentally let her cat get into the washing machine, would have been willing to kidnap an eighty year old woman just so they could make the Blue Patches sequel perfect for Donnie, and refers to said movie get destroyed as it getting "killed".
Homestuck's Jade Harley is pretty much constantly upbeat and energetic, and is her friends all seem to think she is a normal, if slightly excitable, girl. She lives on an uncharted island in the somewhere in the Pacific, which was discovered by her grandfather, a business tycoon and noted polymath. After his death, she taxidermied his body, as is the Harley family tradition, and mounted his corpse in the foyer. She is a narcoleptic, and frequently has visions in her sleep. She is currently cared for by Becquerel, her Big FriendlyBadassGod Dog.
Any of the female trolls would qualify as a Strange Girl were it not for how weird all of the trolls are and how TroperifficHomestuck is in general, but Aradia and Vriska stand out. Aradia spends most of the story as an EmotionlessStringy-Haired Ghost Girl who sets the trolls' side of the plot in motion because the voices of the dead told her to. Vriska's a filthy-rich serial killer with a fixation on spiders and a Complexity Addiction who manages to scare most of the other trolls despite them being a race of bloodthirsty Jerkasses. The two of them also have the strongestPsychic Powers of all the trolls, not to mention being the only two trolls to make it to god tier.
There's also Kanaya, who is extremely strange within Troll society. Not only can she stand sunlight, but she actually enjoys it; she is one of the only trolls that actually cares about fashion and even makes her own clothes; she famously wields a chainsaw disguised as a tube of lipstick. Even her blood is strange, as her particular shade of green is probably rarer than any other troll in the cast save Feferi and Karkat. As it turns out, her role in the trolls' session is analogous to Jade's, so being a strange girl might be a qualifying characteristic for Heroes of Space. She's actually a Rainbow Drinker, which explains a lot of the strangeness.
Jade Sinclair (Generator) of the Whateley Universe. Petite, of Japanese descent, and looking only about ten even though she is fourteen or so. She is also Jinn Sinclair (it's complicated) who claims to be Jade's dead older sister. She is also a cabbit and a Hello Kitty compact and a living lawn mower blade. Like I said, it's complicated. Even the other members of Team Kimba think she's wacky.
Her conversations are usually strange, too. She went through a phase where she would randomly give Dieter an object to hold, then take it back some time later or forget that she gave him something in the first place.
Growing Up Creepie has Creepie, an orphan raised by insects. All things considered, her down to earth attitude is either a flying contrast to her upbringing or a completely logical result.
Ruby Gloom fits this trope perfectly, being bright, cheerful and friendly in every situation. The only thing that can get her down is bright colors. The strange thing is that the merchandise on which the show was based kind of gave Ruby a very melancholic feel, but whatever.
Pi is a rare male example from Sabrina: The Animated Series. He always wears a hat that covers his eyes and seems obsessed with aliens and paranormal.
Lilo, whose classmates reject her for being weird...The creepy homemade doll, making voodoo on her hula halau-mates using spoons and pickle jars ("My friends need to be punished"), Elvis obsession, and worship of a fish (who apparently controls the weather) don't help her case. Fortunately, Lilo has good friends in a bunch of aliens who came to Earth. And a hovercraft!
Hell, one could even say that's the main reason none of the islanders give a rats ass about there being a shitload of aliens all over the place. After some of the weird stuff Lilo has gotten into in the past, they are just used to strange shit happening on a regular basis.
Well, that and they're Hawaiians, whether by birth or by locale, and Hawaiians tend to have "hang loose" embedded in their DNA somewhere.
Jinora from The Legend of Korra is normally a Cute Bookworm, but her view of romantic relationships seems to be... special, to say at least. She swoons at the thought of one of her story heroines commiting genocide and suicide over being rejected by her Love Interest.