I know that foreshadowing is when you drop hints about the future, but is there a name for it when a statement which is innocuous at the time suddenly becomes very important in the light of new events?
The underlying principle is Chekhov's gun:
Remove everything that has no relevance to the story. If you say in the first chapter that there is a rifle hanging on the wall, in the second or third chapter it absolutely must go off. If it's not going to be fired, it shouldn't be hanging there. —Anton Chekhov
The moment when the significance becomes apparent is an anagnorisis or ‘recognition’; Aristotle regards as most artistic those plays in which a recognition leads to a climactic peripeteia or reversal in the action.
In Aristotle anagnorisis is a recognition by a character; but Classical drama, grounded in retelling of traditional stories, did not practice concealment from the audience. Modern plays often do, and criticism of modern dramaturgy has extended the term to moments when the audience realizes what is going on.
Prefiguration. Or more vaguely, an adumbration. Both are Biblical. According to some scholars, the Sybelline Oracles adumbrated the Book of Revelations.
is a form of prolepsis or foreshadowing that hints at some looming disaster in which the first-person narrator laments his or her course of action which precipitates some or other unfortunate series of actions. Classically, the narrator never makes explicit the nature of the mistake until both the narrator and the reader have realized the consequence of the error. If done well, this literary device can add suspense or dramatic irony; if overdone, it invites comparison of the story to Victorian melodrama and sub-standard popular fiction.