My own groups are trying out the luck system from Tales of the Valiant and like it so far. It's extra bookkeeping but has some fun options and makes a series of bad rolls work out much better. As a GM, I like it a lot because I don't have to keep track of it at all. It's all player-managed.

See Playtest Packet 2 in the playtest packets on https://www.talesofthevaliant.com

While this is all well and good, I think the OneD&D playtest provided a great fix on the biggest issue I've had with the mechanic: Remembering it exists.

Simply stating that on an unmodified 20 or 1 that Inspiration is granted is a fantastic way to remind the GM that they should give this out more often. It's gotten to the point that in cases where I would have previously given ad-hoc advantage, I now give inspiration where the player can make use of it if they feel like they want or not.

Also I have adopted those rules for both hero points in PF2e, and my regular D&D5e game. I mean both of them.

How it works....

A player rolls a Natural 1 - They get inspiration to use on a future roll. A player rolls a Natural 20 - They choose another player at the table who gets inspiration.

These two together more or less ensures that there's an amount of inspiration floating around all the time. But this also solves an issue where a hot-streak gets hotter, and someone with luck early on will get more of a spotlight later. By making success something that gives a boon to someone else, you build espirt de corps as everyone celebrates eachothers' successes more, and the spotlight be more likely to move to another player.

As I said before, this system works fantastic in both Pathfinder and D&D. The only downside is that it feels like PCs have an extra layer of plot armor, but that is mitigated by the fact that it wasn't my decision that I made capriciously. Also it makes math rocks a little more 'WEEE'!

The Lord of the Rings 5e game has a pool of Inspiration shared by the entire party called the Fellowship Pool (this is in addition to Inspiration for individuals). It can be spent by anyone in the party to roll with Advantage. The pool is set by the number of players, who their Patreon is and certain Feats (called Virtues in LOTR). The pool can also be spent to trigger special events with their Patron.

The points can only be refreshed during the Fellowship phase which is the players downtime between adventures and covers periods of weeks so typically happens 2-4 times per year. It works really well for the Tolkien style of 5e....it is the only 5e game where all 6 of my players have bought the Rulebook!

I have only been at one table that used inspiration which used them additively so you could have 10-15 inspirations. This basically meant that no one could ever fail a check which was a real bummer to play. Hero points like this can be ported over without much issue.

However, I would rather join in the Force Point Mechanics from FFG Star Wars games (it has also been used in other star wares games). Basically you spend light side points to do exactly what you mentioned above, however spending this will create 1 dark side point. These point can be used by the DM to do the exact same thing for monsters. These points are pooled across the table which lets everyone manage them. This is easier to see and others can suggest to spend them as a team when a big fails happens. Its super fun

...last week, i had this brilliant idea for how to use both valiant's luck points and a slightly-modified version of D+D's inspiration system simultaneously, but i hesitated posting to reddit for obvious reasons and now i've totally forgotten how to modify the inspiration mechanic (but it was something other than advantage)...

...which really contributes nothing meaningful to this discussion, i'm just kicking myself for not transcribing the mechanic at the time...

In case you want to play D&D 5E but without supporting WotC while also having more customisation options: Check out Level Up: Advanced 5th Edition. It's also got its own thing for inspiration: during character creation, you pick – other than heritage, background, and culture – a destiny. It's what your character is meant to do, their story, their goals. This destiny gives you another means of gaining inspiration (things tied to your destiny) and a unique way to spend inspiration. (Also, you gain a special destiny feat when you accomplish a big thing related to your destiny!)

Aside from the death save thing, that's really just a minor tweak on the basic mechanic of 5e's inspiration (rerolling after the fact instead of getting advantage beforehand), plus letting them stack when accumulated. The biggest problem with 5e inspiration being hoarded isn't that it doesn't stack, it's that many DMs just don't give it out with any frequency whatsoever even though the system's description encourages DMs to give it out for all the reasons mentioned (cool roleplay, clever tactics, etc). The idea is supposed to be to encourage players to "go ham" getting into character and immersing themselves into the game to earn it back after using it, not to lack such motivation because they feel safe with three do-overs in their back pocket.

The system isn't flawed, it's just underused and the slightly different version of it doesn't change anything unless the DM/GM actually gives it out. And I would think players would be more likely to ause inspiration or hero points or drama dice or epicness metanuggets or whatever you want to call it if they heard the DM say "I would give you inspiration for that but you haven't used the one you already have" than they would if they didn't "waste" that extra one by just tossing it on the pile they already have "in case of emergency" like an offline FPS player hoarding their rocket ammo but never using their most powerful weapon because they might need it all against the next tough enemy.

And I'm not saying either slight variation of the same basic mechanic is necessarily better. I'm saying that the solution the problem of it not getting used will remain the same in both unless the DM/GM takes the exact same actions with either (actually handing it out). Again, the problem isn't the system, it's operator error.

There's enough ways to get advantage in D&D without inspiration that it's no wonder players ignore it–and why we explore variants. I want inspiration to promote shared storytelling at my table. I borrow a bit from Fate and let players spend Fate Points to add aspects to NPCs and locations. Want to make that duchess paranoid so she's more receptive to the party's tale of an Aboleth controlling her children? Go for it. She'll be a more memorable character than whatever I planned because my players are invested.

I offer fate points to players when their characters fail a d20 roll if they accept a story-related complication. The player doesn't have to accept, but I try to make the complication increase excitement at the table. If I can't think of a complication that fits, the game just keeps moving.

I've seen this kind of thing a lot and I always think it's very much like reinventing fate points from fate. But I like Fate a lot, so I'm not mad about it.

The tldr for fate points in: you get some every session. You get more when something about your character complicates things in an interesting way. You can spend them to boost rolls if there's a narrative justification, or to add story details to a scene (ie: there should totally be a trap door, the queen should totally have a pet elephant, etc)

I like this better than vanilla inspiration because it requires players to pay attention to the story. You can't boost your medicine check if nothing about your character is medical, but if you have a Healer background that's an easy sell.

Some people hate that more "writer room" style. Some people just want to sit back and be told a story. They probably won't do well with this.

Anyway fate's srd is free and I highly recommend it.

So, basically it's "Inspiration" crossed with the Chronurgy Wizard's Chronal Shift feature.

I wouldn't add this to 5e (unless it replaces Inspiration) just because it's another resource and mechanic for each player to track. Maybe it makes sense in p2e.

Furthermore, it's annoying enough having one player at the table being able to force rerolls periodically, but having the whole table be able to do this would interrupt flow fairly frequently. At least it only applies to your own rolls.

I very briefly tried (with a table of newbies) to have players award each other inspiration and no one ever did it. I always forget to do it when I am DMing. My thought process is to let every player award one (if people start to abuse it, no need for a limit at the start) inspiration whenever another player has their character do something that excites you as a player. So basically encourage enjoyable roleplaying. It can be whatever you like. If they make you laugh and you want to see more of that then that's a good reward. My thought process is that it gives players a way to communicate and award behaviors they like to see during play as well as encouraging good showmanship and cheering your friends on.

But like I said, I only tried it once and it didn't work well but also everyone was a newbie. If you view rpg players as a spectrum from theater nerds to videogame nerds (not that such a dichotomy exists, people can be in both or neither) then this will probably work more for the theater nerd side of the spectrum. But it still feels like a good flexible system because if the crunchier folks want to reward things like an expertly placed AoE spell then that is fine too because if that's what the table likes to see them they reward it.

Also it can be used for players to cheer each other up. Like if someone keeps rolling a lot of misses someone could give them an inspiration if they seem it worthy (especially if they roleplay the failures in enjoyable ways).

Honestly it sounds a lot like the Heroic Inspiration system from the OneDnD playtest material. Mostly the part about getting one every session (human characters get one for free every long rest) and having more defined reasons for a DM to hand it out.