Obfuscating Stupidity/Fan Works

During the late '90s, Usagi of Sailor Moon practically ruled this trope. It was all the rage for fanfic writers to make Usagi into a God Mode Sue who was only pretending to be an idiot, often by leaving the Inners and dropping the act. That she still can't write proper hiragana as queen and routinely swipes her kid's candy in the canon is conveniently ignored.
- She is The Ditz in the anime, but shows that she is actually pretty smart in the manga, especially as it goes on and the burdens of leadership take their toll. Venus can be considered another case as well, since it is hinted that she speaks fluent English, a language that is just as hard for the Japanese to learn as their own language is for native English speakers. (They're in completely different language families, so have no common baselines).
In Detective Conan, a few fanfic authors like to play with the idea that Richard Moore is aware of what Conan is doing, or faking being asleep. Has a basis in the simple fact that being injected countless times with the same sleeping drug will eventually result in the body resisting it.
Crabbe and Goyle of Harry Potter are quite commonly described as doing this. Sometimes they are just shy or quiet, and that combined with their appearance, their love for food and their status as Draco Malfoy's cronies make people think they are stupid. Other times they are actively pretending to be stupid so that people won't expect anything of them. In both cases they either actually like Draco and he likes them, or they protect Draco physically (as is implied in canon) and he protects them socially. In one particular fic, they weren't stupid, but since they suffered from undiagnosed dyslexia, everyone thought they were.
Kyon, in Kyon: Big Damn Hero. He is shown to be highly fluent in English, easily beating the brigade in Scrabble, and invokes Gratuitous English. He also learns German from Tsuruya extremely quickly. Aside from language skills, Kyon also managed to convince a renegade time traveler to pull off a Stable Time Loop for him, or risk being killed, because Kyon can go back in time to take his place, thus rendering his Contractual Immortality moot.
The Evangelion fanfic Matters of Faith has Rei doing SCIENCE without anyone knowing. When one of Shinji's innocent comments back her against a wall in a conversation, she suspects him of doing this (with the justification that it's probably Gendo's genes in play).
Italy, of all people, in Heta Oni.
Light in My Stupid Reality has faked being Book Dumb all his life in order to protect himself from L. He has it down to an art:

Light: One does not just randomly select a question or 2 and try to get them wrong. No. Nor do you do it to every 4th question or some other such ridiculous, predictable pattern like that. The key lies in, like all things, strategy. You need to choose ahead of time a subject or 2 your awful at, a few your neutral, and 1 your good at. Then whenever you get homework, consistently get the same basic principals wrong, and get it wrong on the test as well. Also, you have to make a show of not paying attention in class, or else suspicions will be raised, and you'll be given a special "learning disability," and assigned a tutor pronto.