My favorite house rule is one that is so popular, it forms the basis for many current RPG systems. The fate point, hero point, benny, stitch, or similar abstract "get out of a tight spot" mechanic that rewards players for role-playing well, and gives them an opportunity to control the dice a little bit more than they otherwise might.
Back in my AD&D 2nd Edition days, I used hero cards, a set of homemade index cards with little bonuses written on them. If you RPed well (or brought snacks), you got a card that had a circumstantial bonus, like "you take no damage from this fall" or "you have the upper hand in this grapple." The idea what that you could save your hero cards to use one time, at some appropriate later time. The execution was weak– nobody liked having to keep track of cards from session to session. But it was one of my better ideas for a basic, circumstantial benefit for players. If I were to make something similar today, the cards would be class-specific, and you would simply be able to draw one from your deck when the GM wanted to reward you (like having Inspiration in 5e). And your deck would be shuffled and you’d have no cards drawn at the start of each session.
Come to think of it, this would make an interesting deck-builder type of thing for RPGs…. No. No, let me put that aside. I have enough project going as it is….