Squid Girl/WMG

Ika Musume is a Puppeteer Parasite.
The "squid hat" is the real Ika Musume.

When she decided to invade the surface she realized that she would need a body that could function on land, and to that end she attacked some poor, anonymous swimmer and latched onto them headcrab style. This would explain a number of oddities within the series, all of which have been pointed out as unusual by the characters themselves:
 * It's not that Ika would die if her "hat" is removed, it's that Ika cannot survive out of the water without a host.
 * She is likely extracting oxygen directly from the host's bloodstream, which allows her to breathe air; without a host, she'll suffocate if not returned to the water immediately. She can't breathe underwater because this supersedes her own respiratory mechanisms; she relies on her host for nourishment so long as she's attached to it.
 * Also explains her eating habits. Shrimp, lots of it, crammed into her (or rather, her host's) mouth.
 * Ika mistook Takeru for a comrade of hers when he was wearing a squid-hat not because she failed to recognize him with the hat on, but because she assumed that he'd been recently infested by one of her people. When she addressed him as a comrade, she was talking to the squid. And when she encountered Takeru's empty hat, she directed her grief at the hat, not at Takeru's hatless body.
 * Ika does not know how to swim...with a human body. Without one she'd jet about as a squid usually would, but having a hundred-odd pounds of human weighing her down makes this rather more difficult.
 * Ika's Easy Amnesia from falling down the stairs is a result of the impact jarring the interface between Ika and her host's brain. Attempting to remove her "hat" stressed the connection and allowed it to realign.
 * And, finally and most damningly, she looks like a squid sitting on a girl's head.


 * This troper doesn't think so. While the Puppeteer Parasite idea explains the hat and super powered tentacles, it doesn't explain her ability to hold her breath a long time, make her body glow (not just the tentacles), and spit squid ink.
 * Just make some biological modifications. If that organism can link neural systems, surely a bit of in-vivo gene-tweaking is not beyond its capabilities. Also supporting this theory is that right after she came to shore, she introduced herself in Japanese, despite never having had the opportunity to learn the language. I think this show is about to take a turn for the darker...
 * Ika has literally bonded with her host body, which allows her to modify it as she chooses, but at the cost of removal = death. If she were to jettison her host body, she could probablt do so, but it would be a long and complex process, ending in the host's death. Or it could truly be permanent, and Ika is one of the few of her species to ever go through with this process (and also explains why we don't see more of Ika's kind.)
 * She speaks Japanese because of Translation Convention, I guess.

Ika Musume is a sea goddess.
One of the Japanese myriad Gods (Kami). It's just that she doesn't know she is one. She naturally gathers the attention and affections of those around her (faith), the particular beach she's staying at is clearly a lot safer with her around, and she even seems to bring varying amounts of luck to those around her.

Ika misunderstood the mission for going to the surface in the first place.
Ika is the only squid person we see in the show. Who's to say that taking over the world isn't what others instructed her to do? As naive and easily-distracted as she is, she misinterpreted what her original purpose was for going on land. The pollution reason could very well have formed in her head.
 * How very Invader Zim.

Ika is Obfuscating Stupidity without realizing it.
Inside Ika is something that really is your standard gigantic, evil, Lovecraftian Eldritch Abomination, and she only needs a stimulus to wake it up so she can realistically take over the world! It will reveal itself in the anime finale.

Ika is the result of a tentacle monster and a school girl
... nuff' said
 * Alternatively, She is a female tentacle monster. It would explain why males are so attracted to female humans, for the reason that female humans heavily resemble female tentacle monsters ( minus the Prehensile Hair)

Ika is the daughter of the great Cthulhu

 * Comes from the depth of the sea? Check!
 * Has a humanoid body? Check!
 * Has tentacles growing from her head? Check!
 * Wants to overthrow humanity? Check!
 * Is 50 feet high? ... well, she's still growing...
 * Ahem.
 * AHEM
 * AHEM AHEM
 * To be specify, she is Cthylla. With her guadians become dimmish in number thank to polution, the three eyes winged colossal squid she once take just no longer suitable. So she take Moe human form with adorable personality, then blend into humans society to ensure her survival.

The show is a prequel to Frank Schaetzings "The Swarm"
Ika Musume is the first attempt of the Yrr to get rid of humanity by sending a genetically engineered organism on shore. It won't work out so well, so they will decide to go for murderous whales and crustances filled with biological weapons instead.

Pre-brainwashed Ika was Yotsuba!
Expanding on the hat parasite theory.

The siblings' parents are dead.
Chizuru is the apparently the oldest sibling of the three (having evidently graduated from school), but yet does not look all that much older than middle child Eiko. However, Eiko is the only one of the two who is shown to be going to high school, which is neither compulsory nor free in Japan. The Lemon Beach house, while fairly successful, seems to be their only source of income which means that it would be difficult (maybe disastrous) for both girls to go to school and run the place long enough to make enough money to go through the month. Takeru is an elementary school kid, so he can't run the place by himself, and no parents around means they can't run it. Since squid die after they mate, Ika Musume has no concept of parents and so doesn't question the fact that they apparently don't have any around. Gorou may not see any reason to question anything since he and Eiko are childhood friends. Sanae is their next-door neighbor and another one of Eiko's friends so she feels no need to question anything either. As for Nagisa, there's no explanation for her not to question their lack of parents other than not doing so out of politeness.
 * And then there's the flashback scene in the episode where they visit the amusement park: Eiko remembers her parents and starts crying.

Chizuru from Negima and Chizuru from Ika have common ancestor.
Which they are obviously named after. If said ancestor is still alive - she is scariest grandma EVER!

This series takes place in the same universe as Touhou Project
Ika is a squid youkai. She even has a hat!
 * Nice Hat? Check! Takes the form of a cute girl? Check! A bit of an idiot? Check! I'd say it's very likely that Ika is a squid youkai!
 * As the opening of Season 2 she can glide if not fly just like a sea gull

Ika Musume is the daughter of Ikadevil.

 * Which is why she cheered for his expy in the stage show episode.

Ika Musume was sent to "invade" the surface for her own good
Whatever she really is she's more like a human than a squid, and the others felt she would be happier among humans. They called it an invasion so that it wouldn't seem like they were kicking her out.

Mini-Ika is real.
The first time we're introduced to Mini-Ika, it's revealed to be Sanae's Dream. However, in season 2 we're treated to a second adventure of Mini-Ika, seemingly within the same framework of the original dream. Except this time it was Gorou's dream, not Sanae's. Both of them are perceiving a parallel world in which Mini-Ika truly exists.
 * Also, one of the two direct-to-video Mini-Ika stories is not presented as somebody's dream.