Not sure I could pick just one “favorite” revolutionary mechanic, since I’m constantly discovering them. I won’t talk about fan mail, since I just did. I’m gonna skip character questionnaires because that’s tooting my own horn, and they’re most useful in a particular playstyle.
But a mechanic that was revolutionary, broadly applicable, and continues to be explored, is the multi-axis die roll.
A multi-axis die roll is when a single roll of the dice† gives you multiple, distinct answers. In most RPGs, rolling the dice tells you how well you succeed, possibly extending that to determining the effect of the action (such as how much damage an attack does, or where it hits). But a multi-axis roll takes into account multiple things on more-or-less independent axes, at a minimum telling you both whether or not you succeeded and why.
I believe that the first published game to use them was Immortal: the Invisible War. Though those rules are so poorly written that I’ll forgive you if you didn’t recognize them as such. In that game, for each obstacle you had to overcome, you rolled a separate stat, but all simultaneously. In a typical RPG, hitting someone might be a difficulty of 10, but hitting them in poor light might require a 15. In Immortal, when you want to hit someone you roll your combat stat against that 10 difficulty. If they’re in the dark, you also roll your perception stat, against a 5 difficulty. You have to succeed at both to succeed at the task, and if you don’t, you know which part you failed at.
I think Four Colors al Fresco was the next game to use this concept, though when we first came up with it we didn’t even realize that’s what we were doing. When you roll the dice, you’re sorting out which forces are more or less influential—this tells you not only if your character succeeded at a task, but why the succeeded. And instead of numerical degrees of success, it tells you kinds of success. Did you succeed because of your capabilities, or because of your opponent’s lack of capabilities, or because of circumstances?
Bacchanal used a slightly different, cleverer implementation. It completely replaces character stats with just the dice, and the dice determine both what happens and why.
The next game to use this mechanism is Don’t Rest Your Head. This is more like Immortal, in that you have different dice to represent different challenges and capabilities, but the results are more like in al Fresco or Bacchanal. In addition to telling you success or failure, which die leads to that result has in impact, and the other dice in your dice pool also tell you other things, like stress or consequences.
Finally, the most recent game to use a variation on this mechanic uses special dice: the new Star Wars games (Edge of the Empire, etc.) have separate dice for each aspect of the roll, combining the multi-axis results of Don’t Rest Your Head and Four Colors al Fresco with the multiple character dice of Immortal. And by using special dice, it’s easy to quickly identify which is which, thus streamlining the one awkward part of multi-axis die rolls.
Interestingly, I believe that every one of these games independently came to the same mechanism. I know that Bacchanal was created without knowledge of Four Colors al Fresco. And from a combination of conversations and noting the differences, it doesn’t look like any of the others were aware of the earlier games. I look forward to seeing what others do with this technique, now that it has been in as well-known of a game as the new Star Wars.
† There might be a way to do it with a single die, but it’s not obvious to me—I think you need to be rolling multiple dice to implement this technique.