Hermione constantly saves the day in Harry Potter because she was the only one paying attention in class when the relevant magic was discussed.
This sort of thing is even done across books. In the fourth book, Dumbledore casually mentions a room filled with chamber pots that he found when he desperately needed a bathroom, and then was unable to ever find again. In book 5, the room is formally introduced as the Room of Requirement, and ends up being an integral part of the story in every remaining book in the series.
And in the very first book, Snape quizzes Harry, and one of the questions is about bezoars. In book six, it's used to save Ron's life, although Harry re-learns about it from Snape in a roundabout way.
In Dan Brown's Angels and Demons, an early discussion of how much a small flat object can slow down a falling person saves Robert Langdon's life when he uses a window cover to slow his fall from an exploded helicopter. This is also lampshaded at the time the discussion takes place.
Modern pulp author Matthew Reilly uses this to a ridiculous degree. Two examples:
In Temple, there's a throwaway sentence from the protagonist about how he'll need to change his PIN number after reading a story in the paper about how most people use their birth dates as pass codes. Guess how he defuses the superweapon his brother worked on?
Guess again, it wasn't his birthday. But that example was used as a starting point. His brother always used Elvis' army serial number as his PIN. The Nazi scientist used his supposed date of execution.
In Area 7, a precocious youth found in the middle of a government base delivers a buttload of the kind of trivia kids that age accumulate and share at any opportunity, including how komodo dragons are sensitive to changes in the Earth's magnetic field. So there's a scene where the main character has to fight off komodo dragons in a watery pit with his magnetic grappling hook.
Arthur: You know, it's times like this I wish I'd listened to my mother.
Ford: Why, what did she say?
Arthur: I don't know, I wasn't listening!
In the Strange Matter book 'Knightmare', the protagonist remembers a comic his friend told him about in where crossbow being the only weapon to pierce the bad guy's armor. He later uses a crossbow to evil the pierce the evil knight's armor and kill him.
An opening scene of Galaxy of Fear: Eaten Alive has a zoology lesson mentioning how the fearsome rancor, which usually kills anything it sees... except for the tiny gibbit birds which venture into its mouth to clean its teeth. The characters end up on the planet D'vouran, which is a carnivorous entity that will slowly eat any beings on its surface except for the native Enzeen, who make people want to stay there.
In The Nightmare Machine, Lando teaches Zak how to play cards, and that "A good rule to follow is that if the other guy is acting normal under unusual circumstances, you can bet he's bluffing." Ultimately that's how he knows he's in an illusion later.
The Doomsday Ship has Tash finding a bit of Jedi lore about how sometimes it's better to take "action through inaction", or that waiting is sometimes a valid response. She shows it to her brother. After that, Zak finds his uncle playing dejarik on the computer and taking his time despite an icon on the screen flashing "YOUR MOVE". Uncle Hoole says sometimes the computer has to wait, he's strategizing. This is important later.
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