Drawing Straws
In TTRPGs, you often roll dice, sometimes draw cards, or occasionally pull bricks from a Jenga tower, if you find yourself needing to randomize an outcome. But to my knowledge, there are no games that use one of the oldest methods of all: drawing straws. If anyone knows of any, I'd be very interested in hearing about it.
There are three great things about drawing straws. One is that is not probabilistically or psycologically quite equivalent to rolling dice, as I will go into at great length below. The second is that it's very accessible. In a pinch, you could literally use long blades of grass or something. Or you could use plastic straws, or toothpicks, or q-tips, or pieces of thread, or tear up scraps of paper and mark some of them. As long as you know what rules you're going by, you could play spontaneously wherever, like say if you are trapped on a lifeboat at sea with no rescue in sight. And three, speaking of which, it's inherently kind of ominous. It has been well established, at least in English speaking culture, that drawing the short straw is bad.
There are two main not so great things about this method of randomization. One is that it assumes you're playing in person, which probably most people aren't doing these days. However, we've solved this problem for dice: there just hasn't been demand for a computerized straw-drawer yet. There's a small chance I may attempt to make one myself. The other is that it's slower to set up and carry out. However, that just means you probably want to do it a lot less often than you might with dice, and there's also more of a sense of anticipation, so it probably leads to a particular style of game: one where you build up to a crisis point.
I might make such a game, but for now I'm going to just dump my thoughts into a blog. I'd love it if someone decided to run with one of these. If this directly inspires you to make something, I'd love if you told me about it, and also put me in the acknowledgements if you publish it.
The Basic Case
This is the traditional way of doing this. You have a bunch of straws or other similar objects where one is shorter than the others, one for each person. Someone holds all the straws in a way so that they all appear to be about the same length. Everyone takes one (the person holding the straws is left with the one not picked). Then, you see who ended up with the short straw. Traditionally, ending up with the short straw is bad.
How does this work in a game? Let's say you reach a crisis in the game and it's time to resolve it. Instead of rolling to see what the consequence is, or how bad it is, you draw straws to see who it falls on. This gets interesting when particular consequences will be more serious for some people than others. Maybe someone is already badly injured, or only some people are actively wanted by the law.
Different numbers of straws and people
You could have one person draw once, instead. You can simulate die rolls, in fact. Instead of rolling a 1d6 where a 6 is a success, you draw straws where one is short. If 3-6 are successes, you can have two short straws and a long straw is a success.
You can make this more interesting by having the probability of success be a secret. If you don't know how dangerous something is, that changes the tone and experience of the game. You could even have a 100% success rate or 100% failure rate without the players knowing. In more traditional RPGs which are highly tactical, you wouldn't want to do this, but in a game where everything culminates in a single draw or die roll, with every decision in the previous scene adding a good or bad straw, it might work.
You could also have the option of more draws. You drew a long straw? Maybe you can push your luck and keep going. Or you could have more people than straws. If you go around the circle and everyone draws a long straw, you're good. As soon as someone gets the bad straw, you stop, but the tension ratchets up as you go around the circle.
You know the joke about "using up" the good or bad rolls? You can do this for real with straws. This would be slightly annoying to set up for the first time, but say you have 25 long straws and 5 short straws. Then the GM draws six at random and you draw those. If things went really badly at the beginning you at least know your sacrifices set yourself up for success at the end. If things went well at the beginning, that's maybe worse...
The players determine the straws
One simple approach would be that status effects add bad straws. The effect will be that over time things get more desperate. This would be great for games where the expected outcome is that everything goes terribly wrong and everyone dies in the end and you're playing to find out exactly how.
You could also have stats that add good or bad straws. If a character's strengths are relevant, that adds a good straw, if their weaknesses are relevant, that adds a bad straw.
Or each player could have a pool of good and bad straws. You can decide what to put in, but the more good straws you put in now, the fewer you have later. This also can add time pressure: maybe you think you can resolve it quickly and decisively at the beginning, but if you squander your early good straws, you're in trouble.
Remember, again, that straws could be secret. You could pass the DM the straws without others seeing, perhaps. So maybe it's the zombie apocalypse and you're the one who hasn't told anyone you've been bitten, but as soon as someone drew the short straw, we know someone has been bitten.
For strengths and weaknesses it might be interesting if you don't actually know all of everyone else's strengths and weaknesses before you decide the course of action. The weaknesses could also be something more interesting than "I'm bad at X". Maybe the villain has some leverage over everyone in the party, represented by a short straw. If you still have your short straw at the end you're in troube. If you throw in your short straw and it gets drawn, you explain how you were compromised. But if it doesn't get drawn, you're good. How do you know if the straw was your straw? Maybe they're different colours - or maybe everyone who put a short straw in for that round has their consequence triggered if even just one is drawn.
There's also an obvious Battlestar Galactica board game hidden traitor mechanic here: some of you could be working against the party (or you're all working against each other, Paranoia style). But what might be more interesting in an RPG is if you know who the traitors are out of character, and they are torn between loyalty to the party and betrayal/turning into a werewolf/whatever. Now it's a subset of the players who have straws to get rid of, similar to the above paragraph. If they end the game with no short straws and none having been drawn, they switch sides and join the party for real at the end. Your group would have to be quite comfortable separating in-character and out of character knowledge, and as a player you'd have to be willing to play a character who wants the opposite of what you want, but I feel like it might lead to some interesting stories about split loyalties.