Beam Me Up, Scotty!: The infamous "With great power comes great responsibility" quote from one of the early issues is usually attributed to Uncle Ben. While it wasn't long before it was retconned to be from him, the actual first appearance of the quote was in a narration box.
Not only that, the exact quote is "with great power, there must also come great responsibility". Like matata hakuna, this was likely changed so that it was more marketable as a motto.
It wasn't even a motto in the original comics until somewhere in the 1990's. It was just a phrase that appeared once, but, through the film, became so well-known that Peter has taken to quoting it.
The 1990's cartoon ended up DRILLING this into our heads. By contrast, in The Spectacular Spider-Man we went through practically a whole season before it showed up.
Creator's Favorite: Jim Starlin did not create Spider-Man, but he's expressed fondness of him and inserts him into most of his cosmic based stories, even though Spidey is generally considered a "street-level" hero.
During Marvel's 70th anniversary, it's revealed that Ann Weying (not Eddie Brock) was supposed to be Venom, but Jim Shooter didn't think a woman was "dangerous enough" to go toe-to-toe with Spider-Man. The original She-Venom costume was recycled for Earth X's storylines, with May Parker's variation of Venom resembling a Black Widow spider.
Actually, David Michelinie, who created Venom, simply stated in interviews that Venom was meant to be a woman, who lost her husband and baby during a Spider-Man related incident. There was no mention of Venom being Ann Weying (Eddie's ex-wife).
One of the better examples is the wedding of Mary Jane and Peter Parker, done because Stan Lee wanted to marry them in the daily newspaper comic strip. Of course, that didn't stop Tom Brevoort from claiming that One More Day was actually a "fix" to this form of "meddling".
Science Marches On: Now that we know more about the dangers of radiation, modern versions of the story typically have the spider be genetically engineered rather than radioactive. The radiation, still the source of his powers in the main continuity (well, that or magic), has caused some complications as well. For example, it caused some complications for Mary Jane when she became pregnant with his kid. At one point, Aunt May needed a blood transfusion, and Peter donated his blood... only for his aunt to become incredibly ill from it soon after, because he hadn't taken into account the fact that his blood is radioactive.
What Could Have Been: Stan Lee has gone on record as saying if Gwen Stacy hadn't been killed off, most like she would've ended up married to Peter. Some stories, like House Of M, have toyed with this idea.
Joe Quesada wanted to bring Gwen back from the dead in Brand New Day but it was decided against.
Young Allies offered Anya Corazon a potential chance to shine. It had a cast with the potential to create a very interesting character dynamics, and which Anya shone through as the most down to Earth, and easiest to relate to character. But it was killed by poorly written out enemies, an uninteresting and almost generic story arc, and a refusal by Sean McKeever to give the team any sort of cohesion in spite of their common purpose and motivations.
It really seemed the team was finally coming together by the 5th issue or so, but by then the series was canceled. The Young Allies have been popping up here and there however, most recently in Spider-Island.
Writing by the Seat of Your Pants: During Danny Fingeroth's time as editor, he came up with an ongoing plotline of Spider-Man's parents being alive all along (which began in ASM #365 and ended in ASM #388). However, he gave the story to writer David Michelinie, and couldn't or wouldn't tell him where the plotline was going. Fingeroth didn't even know if the parents were real or fake in some way. Michelinie had to treat the situation ambiguously until it was finally decided that they were robot spies all along.
The Franchise holds a Guinness World Record for having Spider-Man appear in more video games than any other comic book character.
Serendipity Writes the Plot: Technical difficulties meant that they had to render the city in as less a difficult a way as possible. So they explained in-story that New York got covered in fog by a machine made by Doctor Octopus.