Writer-Induced Fanon

Let's say that a show strongly hints at the possibility of Susie having lost a baby as a teenager. Almost all of the fans accept this, but the hints are vague enough so that they can also be interpreted to support the idea that the baby that died was Susie's younger sister. Confirmation for the supporters of the first theory would come in the form of Susie over-empathizing with a mother who has just lost her baby and being very tight-lipped when Joe asks her why (why would be tight lipped if it was her sister?) and getting teary-eyed when watching mothers interact with their children. So while Susie losing her sister fits with the hints (though not as well as the more widely accepted story), it doesn't explain either of these situations, where Susie losing her daughter does.

Of course, there are alternate explanations for both of these situations, but really only one that explains them both, and the show has already hinted at it repeatedly.

This is where writers want to include an idea—any element contributing to the plot or a character—in canon, but don't want to explicitely state it. As a work-around, the writers hint at the idea until it's accepted by most of the fandom.

These hints are usually strong enough that most of the fandom gets the right idea fairly quickly, but not so strong that they can't be ignored or attributed to something else if the viewer dislikes the idea being hinted at. The point is, after all, getting the idea across to a large portion of the fandom while offering a less convenient, but still plausible, alternate explanation for those viewers who don't want to believe.

This tactic is most often employed when writers want to include an element such as mental illness, rape, in older shows, homosexuality, or another sensitive topic in a plotline or a character's backstory as a means of plot or character development, but don't want to explicitly state it to avoid controversy or alienating certain viewers.

If the hints are particularly weak or ambiguous, it can be difficult to determine if they are deliberate or merely a coincidence. Short of a Word of God confirmation of intent, the easiest way to establish this is if the idea is built on as if it was canon. Bear in mind also that, for this trope to apply, the fanon has to result from the hints. If the fanon existed with more than a few supporters before the idea was hinted at it's just fanon.

This trope is most common in live TV shows, but shows up in other mediums fairly often as well.

Can result from intentional Canon Fodder. Contrast with Wild Mass Guessing.

Anime & Manga

 * Ken Akamatsu's Mahou Sensei Negima, Love Hina, and A.I. Love You are heavily implied to be set in the same Verse. He can't outright state this because of legal issues.
 * It is hinted several times in Slayers that Naga is Amelia's older sister.
 * The Striaton Trio from Pokémon Special really being the mysterious Shadow Triad of Team Plasma, due to the overwhelming hints. The fact that it's a common theory for the games and anime helps.
 * At no point is it ever actually stated that Haruhi Suzumiya is set in the author's hometown of Nishinomiya. It is just very strongly implied.

Film

 * Jude Law and Robert Downey, Jr. discussed at length the Ho Yay between Sherlock Holmes and John Watson in interviews to the point of leading several groups into believing that the gay subtext between the characters would actually become text within the film. The actual film portrayal is a fairly straight forward Bromance between two Heterosexual Life Partners who both have female love interests.

Literature

 * The Neverending Story was basically written with the idea that its readers would write their own stories.

Live Action TV

 * From NCIS, Ziva having been raped in Somalia. It was hinted at by Gibbs and Vance throughout the beginning of the seventh season and is widely accepted fanon.
 * The idea is further used throughout the seventh and eighth seasons to help develop Ziva as a character, as well as her relationships with other characters, most substantially Gibbs, Tony, and her father, but remains non-canon.
 * Renee Walker having been raped on Twenty Four. While it was pretty evident that she was raped by Vladimir Laitanan during the events of Day 8, the writers never more than hinted at the idea that she was also raped when she was undercover with the Russians before. This is one of the most widely accepted pieces of fanon in the 24 fandom, many fans even considering it canon.
 * This idea is later used to help validate what Renee ends up doing to Laitanan.
 * From Bones, Brennan having Asperger's Syndrome was an example of this until she was given a Word of God diagnosis. It's still an example of this if you don't consider the Word of God to be canonical.

Web Comic

 * In Order of the Stick, Belkar Bitterleaf's asked if he caused the death of Roy, Miko, Miko's stupid horse, or the oracle, and got a "yes" answer. When Roy fell several hundred feet onto the ground to his death, many fans assumed this fulfilled the requirements, because without Belkar, Roy would not have been able to engage Xykon for that particular fight and thus would not have died. Later it turned out that the prophecy was less ambiguous as when told this by the Oracle, Belkar doesn't buy it and promptly stabs him to death. The reason why it's here is because in the compilation comic, Rich Burlew says he deliberately set this trope up so he could subvert it.

Western Animation

 * In Avatar: The Last Airbender, Zuko's mother was banished for committing "treasonous acts", however it's never stated what these acts were. Coincidentally, she was banished right around the time Fire Lord Azulon died. It was widely believed amongst the fanbase that she was responsible for his death.
 * Someone on the fan list for Adventures of the Galaxy Rangers (where several of the show's writers lurk) brought up the disturbing possibility that the Queen didn't put Zachary in the Psychocrypt right after his capture, but decided to take out some frustrations on him first, especially since the scene where she's standing over his unconscious body was dripping with some disturbing Foe Yay. One of the writers delurked to admit that the writers themselves had very off-color speculations about Her Majesty's sex life. It's about a 50-50 split in the fandom whether she "just" used Mind Rape, or went for something more... inappropriate for a animated show.
 * Adventure Time's creator confirms that Ooo is supposed to be our world After the End when The Magic Comes Back, but the show never states it outright. It does offer hints, though—backgrounds include ruined pieces of society, characters make reference to things from before "the Great Mushroom War," etc. A flashback to Marceline's childhood seems to be set when the disaster is in progress.
 * Bubbline.