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There are a few games that formalise this sort of thing in the rules: Your character is considered to have aspects of their backstory and relationships that have not yet been described. Things that in a traditional story-telling medium might be revealed only at a moment where they become significant to the story.
You can make use of this idea in a game by allowing players to develop extra details of their background at times when they would become appropriate in the game action. They're stuck in Cairo in the 1930s? One of the heroes just happens to have spent time at school with a wealthy Egyptian kid whose parents sent them to the same posh boarding school. They can look them up and bingo - instant local contact! You can either just allow this sort of thing as a matter of course, or you can require some sort of dice roll for it which decides how effective the new background is.
Commentary by memnarch (who has not seen the movie)
Hah; not every source of inspiration needs to be completely positive! And this is probably a funny inversion of the situation too, though it likely also makes more sense here. How would we get any of this information before in the movie? If this was an episodic Disney+ show, I could see Rose and/or Hays Minor getting a mention in scenes prior to the start of the space fight. But also, an info drop here would disrupt the flow of the overall scene. We just had the rest of the bomber squad get blown up like 30 seconds ago in the film!
Anyway, with a scene like this, odd as it is, I think there's a 70% chance this works to get the button to drop. An emotional little aside like this does feel like it'd help play up the importance of success.
Another odd thought I just had; what if the "little sister Rose" is actually Ben's current character Tallisan? I know that the Irregulars have occasionally changed names (pretty sure Kyle Katarn was actually from some of the video games), so that could have happened here as well. I'm not sure if that'd make it a better scene or not, but I think it would make more sense than randomly grabbing a medallion at least and giving the ladder another boot.
Transcript
Paige: So the inertial damper won’t let me take advantage of resonance.
Paige: Can I haul myself up the ladder?
GM: You’ll need a Will save at -10 to overcome your ankle pain. Or...
Paige: Yes?
GM: You could find a character story reason to justify Extra Effort.
Paige: Hmm. I think of my little sister... uh... Rose.
Paige: We’re from Hays Minor in the Otomok system. We have matching medallions made of Haysian smelt gold, that interlock in a yin-yang shape.
Paige: I grasp my medallion and focus my thoughts on her.
GM: Okay. You’re inspired to conquer your pain and climb. Roll at—
Paige: No! That’ll take too long.
Paige: I picture Rose in my mind. I imagine the ladder to be her, and I kick that little brat as damn well hard as I can!
[SFX]: Boot!
GM: Um...
Tallie: Hey, you asked Pete to roleplay, not us.