This is a wide-open question, and historically around here the wide-open questions fall flat and deflate with a faint sad whistling sound. But I'll try it anyway.
What are the archetypes of interactive folk tales and fairy tales? I mean, what are the natural shapes of the things?
We have fairy-tale notions -- and maybe they date back no farther than Grimm and Lang, I'm no researcher, but we have them anyhow -- that if there are three brothers, then the first one gets the title and the second one gets the wealth and the third one gets to be poor and honest and goes off to be a protagonist. Three sisters (or nine, or twelve) are rarely even that lucky. You give a coin to a beggar so that he will turn out to be a wizard or the king of this-or-that; misery follows innocence and leads to triumph; and you always fail after succeeding twice, or succeed after failing twice.
(That last point should probably be tied to the observation that second marriages always work out miserably. I don't know where that one leads.)
But all of this pre-supposes a certain... certainty. Inevitability. These stories come to us in books, and there is a way the story goes. (Even if the movie then re-stitches the whole thing into a hat or a pterodactyl.)
What does a story look like when interactive tools appear, and the constraint of print and performance is removed?
I know, this is the core question of the game-design era, and I'm not going to solve it. But the fairy-tale approach appeals to me, because fairy-tale archetypes give us a model of story ideas that are simple -- boiled-down, even -- and yet still resonant. Surely we can say something as simple as "there were three brothers..." while incorporating player choice.
There were three... brothers? Sisters? Siblings? If the player merely chooses the genders and then lets the story run, is that interactivity? (Yes, and probably interestingly. But this addresses the gender roles of traditional fairy tales, rather than their static-fiction form.) If you choose the character, with his or her particular motivation, and then let that run? (Perhaps.)
There were three siblings, and the first was... The second was... Does the story have to be about the third? Can each sibling have his or her own adventure? (Certainly. This is too simple, though, if you just write three stories and paste them together at the front. The point of three siblings is so that we can cheer the least and unluckiest one to victory. Now, if each protagonist thinks he or she is the least and unluckiest -- because they all value different things -- and then each one sees the others stumbling somehow to failure, and sets off to rescue them, while being rescued along the way in two different and (from the interior view) less crucial ways... I think there's some silver to be mined in that hill.)
A child became lost in a forest, and... what happened next? The child traps or defeats the monster and escapes. (Or is devoured, sure, but that forest path doesn't need my feet to be well-trodden.) But how does it happen? (A cut leaf, a flask of spring water, the words to make the roses grow. Is it unreasonable to offer any of those tasks as the story, and let the player choose which one to unfold? The ending is inevitable, but the middle can go various ways. Or you could flip back and learn what happened before the beginning, when the innocent childhood wasn't so simple. Or it might be the beggar's story, after all, who gave the child a flask in return for...)
We have, to be sure, a set of fairy-tale tropes much like IF puzzles: fetch quests, token-gathering, and riddles. So we have the whole array of IF devices that apply to puzzles. Multiple solutions, optional puzzles, free ordering of puzzles, rewards or story events ordered independently of puzzle order. This is 1990s IF technology, and easy to take for granted, but worth mentioning.
@peterb suggested digging into the layers of retelling -- grandmother may tumble out of the wolf's corpse smiling, or maybe eaten is eaten, if that's what you want of it. Underneath the fairy-tale forest is the Schwarzwald, and below that starving bandits, perhaps. I like that notion.
We might have three stories stitched together more delicately: a cause here, an effect there. The interactivity is in choosing which story to follow, on the coarse level; but really the player must recognize the connections and cohere the fourth, unspoken story.
I'm not coming anywhere close to an archetype here, I admit. I'm listing particular patterns, if not specific game ideas. It may be that whereas condensed story ideas are recognizable, condensed interaction ideas are toys -- not compelling without their details of gameplay. I can tell you that you will decide who to adopt as your wise old mentor, or in what order you will defeat the conspirators, or even what virtue you will discover on the way to the witch's oven. Are these notions intriguing? They've been tried, and successes do not come to mind. Archetypes grow out of the stories we actually perpetuate, I suppose.