A humorous play on words not intended by the speaker.
The accidental pun is exactly what it sounds like. While it has to be unintentional by the speaker in-story, it can be, and generally is, intended by the author. If the original speaker notices their pun after the fact, they will generally respond with "No pun intended," or something similar. If a Pungeon Master is nearby, they will almost always be pointed out. Oops, no pun intended.
Often the closest The Comically Serious will ever come to telling a joke.
Compare Innocent Innuendo, where a double-meaning is invoked by observers but not the speakers and Accidental Innuendo.
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Anime and Manga
Puella Magi Madoka Magica: Mami Tomoe's name "Mami" means Mommy in Spanish and other languages like German. Considering her mentor role it fits her.
In Hidamari Sketch, Sae gives Hiro a poetry collection covered with an embroidered book cover. Shishuu is a homonym for both "poetry collection" and "embroidery", but when Miyako asks if it was a pun, Sae claims it was accidental.
Mr. Daidoji: Who the Hell are you?! Hel: [chuckles]
The title of Neko Kissa is either this or a BilingualStealth Pun. In Japanese, "Neko" = "Cat" and "Kissa" = "Cafe". So, "Cat Cafe". The title could also be translated as being the word "cat" in two different languages - Japanese and Finnish: Neko (Japanese), Kissa (Finnish).
In Little Witch Academia, when a minotaur is approaching antagonist Diana, her friends Hannah and Barbera try to hell her it's behind her (ushiro) but only manage to stammer out ushi (cow).
During the Justice League of America event JLApe, Batman is briefing Nightwing on a Gorilla City crimelord who acts like he's in a '30s gangster film:
Batman: Grimm wouldn't be the first hood to ape old movies. Nightwing: "Ape". Batman: Sorry. Unintentional.
Meta-example in the first chapter of Red Fire, Red Planet. There's a chief petty officer in the chapter named Sally Blackhawk, who the author's notes state is supposed to be of primarily Shoshoni Indian descent. The author's notes also state that, yes, "Chief Blackhawk" sounds like a really bad pun from that angle and that it wasn't intentional.
The Hudsucker Proxy has the hero surrounded by reporters, excited about his invention of the hula hoop. When he comments on all of the hoopla surrounding it, the reporters laugh at his cleverness and he winds up very confused. After it's explained to him, he uses it all the time in future press conferences with far less stellar results.
Sort of a meta-example in A Knight's Tale: When William notices the trademark (an upside-down Nike logo) on the armor that Kate made for him, Kate mentions that she had put it on the armor "in case some other knight should admire (her) work." Neither the script-writers nor the cast were aware at the time that one of Nike's co-founders was named Phil Knight.
In Doom: Hell on Earth, Arlene and Albert are out trying to find computer connectors for their hacker Jill. Arlene tells Albert that they shouldn't have any problems finding the jacks for Jill. She starts giggling when she catches it and Albert takes longer to figure it out.
Live Action Television
One episode of Drake & Josh features the following exchange:
Angry factory lady: Let the sushi roll!
Josh: Haha, I get it. Sushi roll.
Angry factory lady: That wasn't a joke. That was an unfortunate coincidental pairing of words.
In the QI segment about lobster-hypnosis, Stephen Fry commits some accidental mussel/muscle punnage, which is immediately jumped on by Phil Jupitus.
Ian Hislop: Didn't Tutu train in Hull? Angus Deayton: Tutu train? No, I don't think so...
A Super-Match posit on Match Game was "_______ Duty." The cast and crew were looking for other possibilities to fill in the blank ("Do your duty," "Civic duty"), when someone offstage cracked "Howdy duty." Gene Rayburn bought it and then discovered how hard he bit.
The tutorial in LEGO Chess on getting in and out of check features this line:
King: So can you check out — 'scuse me, that wasn't a joke — the spaces I can move to?
In Dragon Age II, a potions shop in the Gallows is low on stock and the owner is looking for Hawke to "remedy that". Snarky!Hawke can point out the pun, at which the owner says it wasn't meant as a pun, but that he should remember it for future use.
Tarkin: Whah do we not just flah arahnd ze planet? Vader: This is a quadrillion Imperial tonne space station. Looks like a moon, moves like a moon, steers like a cow. Tarkin: Ha ha! Ah get zis joke. A steer is lahk a cow! You kill me, Lord Vader.
In Homestuck, Terezi was named after the Azerbajani name for the constellation Libra. However, when her name is correctly pronounced with this in mind (which, contrary to popular belief, would be more along the lines of "ter-REE-see"), it sounds very similar to a feminine version of "Tiresias" - the archetypicalBlind Seer from Sophocles's Oedipus trilogy.
League of Super Evil consists of three characters trying to be supervillains and failing miserably at it. Their team name's acronym is L.O.S.E.
In The Fairly Oddparents, after getting their butts kicked by Timmy, Crocker, Dark Laser, and Foop form the League Of Super Evil Revenge Seekers. They aren't even aware of the acronym.
In the first episode of Gravity Falls, Mabel is propositioned by a bunch of gnomes that want to make her their queen.
Jeff: So what d'ya say? Will you join us in holy matri-gnomey? Matri—matrimony! Blah, I can't talk today.
On This Very Wiki
The word "straight" can come off as this on the Playing With pages for sexuality tropes. Occasionally, it's lampshaded with a link to this page.
Every page for an individual Saints Row game begins with "The [number] of the Saints Row series". E.G., the page for Saints Row 2 begins with "The second of the Saints Row series". The accidental pun occurred when the third game was revealed to be titled Saints Row: The Third (which nominally represents the fact that the gang is called "The Third Street Saints", thus being an intentional pun that led to an accidental one).