Most games have some elements of this. Luck points, pushing rolls, advantages, bennies etc. It is some mechanism that allows the players to re-roll after missing a critical roll/save. Most RPGs have some version of this. I like using Hero points to include doing heroic / extra-ordinary shitte. This usage is more geared toward Pulp / Heroic type games to allow character to perform crazy but entertaining actions. This involves the player more into the action of the game. I get them to describe their heroic act when they spend hero points. Eg. Leaping across to catch a rope, swing over the water, and stick the landing on the other side. Under normal circumstances, that could be like 2 DEX checks at med-high DC level.
I’ve also allowed players to spend Hero points to do things outside of their class or level. Eg. Cryomancer at character level 5 can cast a level 6 spell by spending 6 hero points. 1 hero point per spell caster level needed. They can cast a level 6 spell at character level 11. Level 11 spellcaster needed – level 5 spellcaster current = 6. Can this unbalance the game? Potentially. That’s why you need to manage how many hero points a character may have, how many you give out, etc.
Amazing Adventures from TLG has a good system for Fate points. These Luck/Hero/Fate points can also be used in boosting game mechanics. Eg. Add 1d4 or 1d6 or 1d8 based on character level etc. to attack or save roll. Rerolling dice, automatic critical hit (with possible 20% chance of breaking weapon). It can also be used to give a clue / plot break. Ie. Vision from the Gods. Or save from character bleed out and death. It can be used to do a lot of different things. I have not tried this in super hero games as characters are already super human.
As I mentioned, I like using to get players to be creative about doing crazy stuff. As long as they get creative and descriptive of their action, I’m happy. In one recent game, the whole party was about to get hit by a level 7 lightning bolt which would have certainly result in TPK or near TPK. As such, I allowed to spend Hero Points to get out of the way, but they had to describe exactly what they were doing. In this situation, there was a trap pit in the room. One player stated that their character would jump into the pit and stick a dagger in the wall and hang on. Ok done. Next player stated they would jump into the pit and grab onto the 1st person. 3rd player jumped into the pit and grabbed ankles of second player. By the 5th player, they had formed a human chain to the bottom of the pit. It made it silly but also amusing.
I will occasionally give out Hero points in lieu of some other benefit in game (eg. Magic bonus to attribute) or in lieu of XP for a session. Usually characters should not be running around with more than like 10 Hero points. I think Savage Worlds uses like 3 bennies per game session. Call of Cthulhu – you can push your roll once per check.