The World God Only Knows is the Trope Namer, where Elsie frequently tries to excuse Keima's weird behavior by saying he's been playing too many video games, and a caption in chapter 83 "christens" it as it is now.
Prince of Tennis: Ryoma's excuse for being late: "I was helping a pregnant woman to the hospital."
In Fairy Tail, Happy's answer to questions such as "Why are you blue?", "Why can you fly?", or "How come you can talk?" is always "Because I'm a cat!" cheerfully and likely intentionally ignoring the fact that these questions are due to the fact that he is a cat.
It's especially funny when he uses it as an excuse to bathe with the (human) girls.
When the team needs to learn some exposition from an ancient creature, it outright refuses to tell the story to humans. Happy raises his hand and says he's a cat, and the creature instantly changes his mind and starts telling the story to him.
Conan often blurts out smart observations in a crime scene in front of the grownups. His excuse of knowing this? He watches a lot of television hiding the fact he's a 17-year-old Teen Genius inside a 7-year-old body.
Due to the movies being more action packed then the television series, Shinichi has to come up with an excuse for how a seven-year-old child is able to pull off extreme stunts like piloting a helicopter or using a jet-ski. The answer? His dad taught him in Hawaii.
Shirou of In Flight, every time someone calls him on anything strange about him. Eventually his flock start completing the sentence for him.
Shirou: Legacy of a misspent youth.
In Oh God Not Again, anytime a character asks Harry how he knows seemingly impossible things.
Harry: My psychic scar told me.
In Everworld the characters often pretend to be traveling minstrels as a cover for the fact that they're from an alternate dimension
A project. If Saddam Hussein had said he was doing a project on Kuwait, the Gulf War might never have happened.
In Interesting Times, Rincewind visits the Agatean Empire, where foreigners are routinely executed. He's advised to tell anyone who gets suspicious that he's from Bes Pelargic, an Agatean town looked down on by everyone else for being a bit weird.
Likewise, when the Patrician, Nobby and Colon are undercover as Klatchians in Jingo, Sgt Colon is explained as being from Ur, a town proverbial for stupidity. Every time someone gets suspicious about him (he doesn't know what a minaret is, he's unfamiliar with couscous, he's surprised by a flying carpet), the Patrician deftly turns it into an Ur joke, and everyone laughs and relaxes. Colon did it by accident the first time, when after claiming he spoke Klatchian he's called out on it by a couple of locals who ask where he's from, and while trying to think, he says "Er..."
In the first book of the Song of the Lioness series, Alanna always uses "I fell down" to explain her black eyes, and broken bones, and other conspicuous injuries obviously gotten from fighting. This is Justified in that this is the traditional response for brawling pages.
Live Action TV
In Sliders, whenever the team lands on a new world and has to explain why they don't know what's going on, they use the excuse "We're from Canada." We've hardly ever seen it fail. Although one time they had to pretend to be illegal immigrants from Canada the entire episode, who had snuck south into Mexico for work. (Thanks to the non-existence of America in the middle, and Mexico ending up with California.)
That gag may have been used because the show originally filmed in Vancouver although set in San Francisco.
Buffy the Vampire Slayer mostly averted this, our heroes generally coming up with new, individualized (if implausible) excuses each time. On the other hand, Sunnydale's police department (when not suffering from Sunnydale Syndrome) usually covered up monster attacks by explaining that it was "gangs on PCP."
Used often in Smallville during the earlier seasons, when any questions Clark Kent was asked about his interest in the caves or any Native American symbols that were related to his Kryptonian heritage were met with "It's for a term paper" — to the point where Lex Luthor himself actually lampshades it later.
A variation, the question being "Who are you?" and the answer being "I'm the Doctor," rarely yielding any further questions. There are sometimes further questions (hence the show's name), but very rarely answers, at least none that are directly relevant.
In the "Day of the Doctor" special, a conversation reveals that UNIT's go-to excuse for weird happenings is to say that Derren Browndid it.
On LOST, mentioning Canada is always a clue that the speaker is lying. Fugitive Kate tells Ray she's Canadian. Ethan claims to be from Ontario. The Others believe Bonnie and Greta are on assignment in Canada, not jamming signals in the Looking Glass Station. In "The Other 48 Days," Nathan tells Ana-Lucia he's from Canada, which is probably true, but the audience is meant to think it's a lie because Ethan said the same thing.
Fawlty Towers - Manuel being from Barcelona. It explains his bad English. It fails to explain anything else.
Supernatural has a minor variation: The brothers have fake identities every case (FBI agents, doctors, etc), and stick with that for the entirety of the episode—which makes sense, considering they're stuck in a small town. This has led to truly ridiculous claims, such as managing to get everything for a case by being health inspectors.
Naked Woman: Wh-what are you and that kid doing in the women's showers? Sam: Don't worry, ma'am. I'm with the health department.
One episode of Scrubs had Jordan repeatedly justifying her Jerkass tendencies with "My parents were mean to me". Everyone immediately accepts this as a sufficient explanation. At one point she even says it out loud in response to something she was merely thinking. And of course, it turns out to be a lie anyway.
Quite often the males of Rescue Special Ops use this. Jordan and Chase offer to help two women move into their new apartment in hopes of getting a date out of them, they just happen to be using the Rescue car and use the excuse that they are going to get the tyres checked out. When Vince finds out he ropes the guys into helping him move his own furniture on their weekend off effectively making their plan Failure Is the Only Option as they don't even get the women.
Veronica Mars: "It's a project for health class" seems to be Veronica's go to excuse to slip all kinds of stuff past her dad.
Near the beginning of Final Fantasy X, Tidus learns to say "I got too close to Sin's toxin" (which is known to cause confusion and memory loss) to cover for his ignorance of life in Spira.
Nishimoto: That's someone who lived abroad for you.
In Kubera, Fire God Agni, doing his best to disguise himself, tends to get critically close to revealing his secret to our Idiot Hero Kubera. However, whenever something strange happens, he credits to his jacket. That includes surviving her Megaton Punch and spontaneously combusting.
The Fairly OddParents: A Running Gag response earlier on happens every time Timmy is asked where he got his wished-for stuff from. Namely the quick response "Internet". See also the page quote.
One episode subverts this Trope:
Dad: [seeing that Timmy's room is full of stuff that he hasn't bought for him] "Young man, where did you get all these nice things?" Timmy: "Uh...Internet?" Dad: "And where did you get Internet?" Timmy: "Uh...uh..." Dad: [gasps] "He's stammering! Our son is the Wall-2-Wall-Mart shoplifter!" [screams like a girl] Timmy: "What?! You don't think I stole this stuff, do you?" Dad: [screams like a girl again] Timmy: "That's a yes...but I didn't do it!" Dad: [screams like a girl again then passes out] Mom: "That's it, young man, you know you're not supposed to make your father scream like a girl three times in one day!"
That excuse would later backfire on him when he brought a lie detector to show his parents he wasn't a shoplifter. His father asked where he got the lie detector.
In one episode, while trying to explain why he was suddenly rich, he tried both an inheritance claim, and the usual claim, before settling on "I inherited the Internet!"
Weirdness is turned Up to Eleven when Vicky wants to get married to Chip Skylark. Where does she find a justice of the peace willing to marry a pop idol to his crazed teenage fan against his will? "On the Internet!"... Which implies that you really can get anything and everything on the Internet.
Heck, back when the show first started, there was a promotional webpage showing off all the things people other than timmy claimed they got from the internet, including the $5,000 vase Timmy's parents bought for only $1.
In the "Knifin' Around" episode of Space Ghost Coast to Coast featuring Thom Yorke from Radiohead, Space Ghost takes a break from the interview to discretely reveal to his director, Moltar, the fact that he has illegally copied Radiohead CDs. When Yorke appears to be listening in, Space Ghost covers by saying, "Don't look at me... we're talking about dragons!"
Invader Zim runs through all the popular UFO denials, including GIR as a robot Man in Black claiming that Zim, a squishy green humanoid with red eyes and antennae, is an "experimental government aircraft."
UFO Cult Leader: That's an aircraft? Zim:Fool! What else could I be?!
Archer: When Krieger is caught doing one of his weird science experiments, he'll claim he's not really there and it's a hologram programmed to take his place. Although this is slightly plausible, given that his wife is a hologram.
Weather balloons have bit of notoriety amongst UFO enthusiasts as it's probably the most often used explanation ever for UFO sightings, even against the testimony of people who claimed to have flown next to it.
And it doesn't help that during the Cold War there really was an Air Force project focused on what were basically highly secret weather balloons (it involved detecting hidden nuclear testing), and that the Air Force took a while getting the hang of covering up the project's existence. While the convenient public perception that they were really covering up alien contacts evidently came about by incompetence rather than planning, the result worked... and the connection stuck around.
There's a humorous picture floating around the internet with various aircraft profiles (some real, some imagined, and some from media) and the "official" USAF explanation for them.
A perennial joke among archaeologists is that any artifact you can't identify is a "ritual item."
Biologists have a similar joke, that any external characteristics you can't identify are sexual displays.