I've tried a variety of approaches. When played using RAW, I didn't like it. It was easy for both DM and players to forget and I don't like the idea of the DM rewarding players for role playing. It is up to the players how they want to play the game.
Then I let the players decide when to award inspiration. That didn't help.
I like the fleeting luck mechanic of Dungeon Crawl Classics, but it doesn't work well with 5e's advantage/disadvantage mechanic.
I tried some approaches outlined by The Angry GM:
https://theangrygm.com/take-the-suck-out-of-inspiration/
It was better in terms of how inspiration is rewarded, but it was still often overlooked.
In my current campaign, I am doing something entirely different. I tie inspiration with training using some concepts from an EN World En5ider article:
Master and Apprentice (
En5ider 23).
In my game, you have to spend time and money to level up and you make a skill roles. If you score high enough, you can gain an inspiration. The only other way to gain inspiration is by performing sacred rites as a downtime activity (DMG 129). You can only have one inspiration point at a time.
You can use inspiration to automatically succeed on any skill check, saving throw, or attack involving a skill, attribute, tool set, or weapon you are proficient in. Inspiration can also be saved to automatically bump training success up one tier. If you save your inspiration until your next training, it must be used for training. Inspiration never carried over from one level to another.
This might not make a lot of sense without giving the full context on how training and leveling up in my campaign work, but bottom line is that inspiration come from highly successful training or performing sacred rites and, in addition to using it per RAW, you an also use it to bump up your success when you next train to level up.