Archive Panic: The Other Wiki guesses that they may be the most prolific artists ever in terms of released material, in no small part due to Dial-A-Song. It's almost impossible to own everything they've ever released. Even people who pick up and enjoy every studio album are sometimes surprised by songs they don't know that were never released in an album.
"Istanbul (Not Constantinople)" from Flood was originally a Four Lads song, but most people these days only know the TMBG version.
"Dog on Fire", the instrumental theme song for The Daily Show, was originally written and performed by Bob Mould. However, once Jon Stewart became the host a few years into the show's existence, Mould's version was replaced by one performed by They Might Be Giants, which has remained ever since.
"New York City" is a cover of a song by the all-female twee band Cub, whom the Johns were friends with. The original appears on Cub's 1994 album Come Out, Come Out, which had only been released two years prior to TMBG's cover of the song, which helped the fact that more people remembered the cover rather than the original. Flansburgh mentions in the DVD Commentary for Gigantic that he wishes more people knew this, but their version is so popular it's usually near the end of the show and he doesn't feel "Thanks for coming, by the way this next song is not by us..." sounds right.
"Why Does the Sun Shine? (The Sun Is a Mass of Incandescent Gas)" is a cover of a 1959 song by Tom Glazer. The song was woefully obscure when TMBG covered it and actually features a handful of now-inaccurate facts (notably that the sun is actually made of plasma, not gas). TMBG recorded "Why Does The Sun Really Shine?" a factually accurate new version of the song for their 2009 childrens' album Here Comes Science.
"What Is a Shooting Star?", written by Lou Singer and Hy Zaret (who also wrote the aforementioned "Why Does the Sun Shine").
Critical Dissonance: Flood got some surprisingly harsh reviews when it was released, even from reviewers who praised their first two albums (Rolling Stone, Robert Christgau), either accusing them of getting too slick for their major label debut, or fretting that they didn't seem to be getting more "serious". It's their biggest-selling and most beloved album.
"Ana Ng". The Johns even rhythmically tap their teeth to it in the video.
Epileptic Trees: The band seemingly invites these with their abstract approach to songwriting, as shown in the Gigantic documentary with a debate club analyzing the lyrics of "Particle Man". The band neither confirm nor deny any interpretations of their work.
Ear Worm: Many of these in their songs. Such as "Istanbul Not Constantinople".
Face of the Band: John Linnell and John Flansburgh (the original two members) are the ones that all fans of the band think of, although they have other members backing them up, many of them named Dan. Which of the two Johns is more likely to be considered the face depends on your approach to the band. If you go by albums, Linnell sings more of the iconic hits and album headers. But going by concerts, Flansburgh tends to be more of a showman.
Genius Bonus: In-song references can be both diverse and obscure, including history, literature, and pop culture.
Hilarious in Hindsight: Cartoonist Mark Marek did the animation for the "Istanbul (Not Constantinople)" music video in 1990. Nine years later, the videos for "Doctor Worm" and "Why Does The Sun Shine" aired as part of the final season of KaBlam!, which Marek worked on (he made the Henry and June wraparound segments). He later did the animation for "Underwater Woman" in 2015.
Iron Woobie: Mr. Horrible from "Someone Keeps Moving My Chair" gets continuously abused and/or humiliated by the narrator and his Ugliness Men. Horrible's only complaint is the comically disproportionate song title.
Jerkass Woobie: Johnny from "Can't Keep Johnny Down" is a jerkass with a big me-complex, but the root of this behavior is that he feels as if the entire world is against him, and he's made to suffer for it.
Paranoia Fuel: "Hide Away Folk Family", "Where Your Eyes Don't Go", "Someone Keeps Moving My Chair", "It's Not My Birthday", "Hall Of Heads", "A Self Called Nowhere", "The Bells Are Ringing", "Rat Patrol", "Older", "Ant", "Bastard Wants To Hit Me", "I'm Impressed", "Can't Keep Johnny Down", "Black Ops," "Aaa", "What Did I Do To You?"... That's a song off of each (non-children's) studio album (and at least one compilation). Yeah, it's one of the band's favorite tropes, and at least half of those have serious Lyrical Dissonance. "Someone Keeps Moving My Chair" is downright upbeat.
Periphery Demographic: The band were popular with kids since their early days, even when they were playing ostensibly "adult" Rock music. Come the Turn of the Millennium and their album No!, they started making music specifically for kids and families.
Sampled Up: The saxophone hook in "Number Three" is a loop taken from Lou Monte's "Skinny Lena." The band found it on a record they discovered while cleaning out an abandoned apartment.
Sequel Displacement: By far, even to this day, the most well-known album the band has put out is its third, Flood. This is the album that has "Istanbul (Not Constantinople)", "Particle Man", and "Birdhouse In Your Soul". Since it's the most common gateway into fandom of the band, there's even a Fan Community Nickname for those that bought Flood as their first TMBG album: Floodies.
They Changed It, Now It Sucks: In their early days, the Johns worked by themselves and used double-tracking and drum machines to complete the tracks, and they'd play to recordings live. When they finally decided they needed a backup band, fans were pissed. They got over it eventually, though, and now it's hard for most of them to imagine what it would have been like if that had never happened. Their first band album, John Henry, has a controversial status among fans.
Unintentional Period Piece: Surprisingly "I Can Hear You" off of Factory Showroom, and more than just the fact that it was recorded on a wax cylinder. It references the Viper Car Alarm, the newness of being able to call from an airplane, and the then-brand new super-sizing.