Colbert Bump: The SA Let's Play (here) of it pretty much introduced the game to English audiences, and pretty much created a non-Japanese fanbase for the game. It got to the point where there was much more English related buzz about its sequel before it got released, than the first game which was unknown to English audiences.
The Masked Man has gained the nickname Monoluchador.
For a while, Mukuro Ikusaba was referred to as Corpse Warblade more often than by their actual name. It's not hard to see why.
And then we have Chapter 6's reveal of the world outside the school. The fans promptly called it Monoworld.
The game itself has an avalanche of silly nicknames, caused by the strangeness of its own name. It was a popular meme on Tumblr for a time, but has since died down.
"Baseball Bro" for Leon Kuwata. In general, [Title] boy/girl (such as "Heir Boy" for Togami, "Swimmer Girl" for Asahina) is often used to refer to the characters, as some fans find the character's titles easier to remember than their names.
"Swimboobs" for Asahina. A combo of her title, and the fact that her boobs are rather... big.
I Knew It: In the LP thread, thanks to time between updates and readers bouncing ideas off one another, almost every plot twist has been predicted at some point. Bonus points for when the correct guesses are made and then shot down.
However, some of it might be from people who played the game before or looked at spoilers and are presenting what they know happened as "theories" (which is actually currently a big problem with discussions of the anime).
Perhaps the biggest example is the infamous JUNKOS theory. Basically, someone thought that while it made sense for Sayaka to be a Sacrificial Lamb, it didn't make sense for Junko to be one, given that we learned so little of her character. The theory then goes to say Junko must have a Backup Twin! It became a Running Gag in the LP thread until it was proven right!
Megumi Ogata and Hekiru Shiina are reunited from their stint in Magic Knight Rayearth. Except this time Hikaru is actually an evil mastermind and has to be put down by Eagle/Emeraude (instead of Eagle being an available pair for Hikaru, or Emeraude getting Hikaru and her friends to put her down.
Another role-twisting with Hekiru Shiina also exist with Kappei Yamaguchi, as they worked together in an obscure anime called Edens Bowy as the two main characters/lovers: Elyssis/Shiida and Yorne... Except instead Yorne is now Elyssis 'manservant' or sorts, and the former cold-bloodedly murdered the latter.
What Could Have Been: When first envisioned under the working title Distrust, the game would have featured branching pathways where the player's decisions influenced who lived or died: for instance, one pathway would involve student A's death, while in another route student B would be murdered instead, and student A would help you with the investigation.
Your choices would also influence the Main Character's general attitude, as well as determine whether or not certain characters were actually trustworthy.
In keeping with the school theme, Monokuma was envisioned as an anatomical model.
A later revision had a more serious vibe reminscient of Saw, with a Deliberately Monochrome art style. Instead of a courtroom, the equivalent of the school trials was set in a warehouse. The cast is mostly recognisible; however, their designs were less outlandish overall. An animatic of the first execution can be found here; however, it still contains spoilers for the actual game. Everyone are also named differently, for instance, Kazuo Matsuzaki is pretty much a prototype design of Leon Kuwata, right down to the same execution.
In this edition, Kirigiri, who survives the entire game and is the deuteragonist of the story, is the very first victim.
The Visual Fanbookhas a section that contains drafts for all of the other kids' executions. A quick summary of each of them:
If Ishimaru had been a culprit, he would've been sitting on a vehicle in a gorgeous parade with everyone cheering his election to Prime Minister. But then Monokuma would've appeared as a sniper and killed him in a single shot.
If Fukawa had been a culprit, Monokuma would have thrown her into complete darkness except for Togami standing there. Fukawa would've run towards Togami for a kiss, only for a steamroller to appear and crush her into a thin paper.
If Maizono had been a culprit (for instance, if she successfully killed Kuwata), Monokuma would've placed Maizono on-stage promising a perfect idol singing performance, complete with a meter to "score" her. But just as the meter's about to be full, Monokuma would've destroyed the meter, triggering the "failure" condition and causing a giant mantrap to slam shut and kill Maizono.
If Yamada had been a culprit (for instance, if Celes hadn't killed him during Chapter 3), Monokuma would've placed Yamada between Monokuma and "Buko", the magical girl star of Yamada's favorite show Heretic Angel * Mochi Mochi Princess, and he would've been killed by being caught in the crossfire of two "death beams" at once in a parody of an anime duel.
If Asahina had been a culprit, she would've been placed in a water tank with Monokuma dressed as a magician. No, Asahina would not have been sentenced to a slow-and-agonizing-death-by-drowning in front of everyone, but what would've happened isn't much better: Monokuma would've placed a curtain over the water tank, lifted it back up to reveal Asahina surrounded by sharks, covered the water tank again, and lifted the curtain again only to reveal Asahina completely gone, Eaten Aliveby the sharks.
If Fujisaki had been a culprit, Fujisaki would've been sentenced to run forward in a side-scrolling 8-bit platformer game, but a huge group of pixel sprite Monokumas would've caught up to Fujisaki, and Fujisaki's "sprite" would've eventually "popped" out of existence.
If Ogami had been a culprit (and had killed someone besides herself), Ogami would've been standing in wilderness with multiple space aliens and soldiers appearing for her to fight. She would've successfully kicked them all away at first, but they would've kept appearing until finally Ogami wouldn't have been able to keep up the pace, until finally so many appear that they would've just crushed her to death-by-dogpile-and-sheer-amount.
If Togami had been a culprit, Monokuma would've dropped Togami into a garbage can in a place that looks like Hell, and Monokuma would've dressed like an elementary school student and repeatedly thrown stones at Togami for supposedly being a dirty hobo. Togami would've managed to escape, but would have come out in a cold and snowy place and eventually would've died as a combination of his wounds and exposure to the cold. He's an asshole, yeah, but that's still harsh.
If Hagakure had been a culprit, he would've been placed on a game show where he had to choose from three doors. Door A would've grown hands and feet and fled from him, and Door B would've done the same. That would've left Hagakure with no choice but Door C, which would have grown a mouth and eaten him.