[EDIT: Heroic Inspiration is a reroll of one die. Not a reroll of, say, both d4's in a 2d4+2 healing potion.]
Inspiration is great. In vanilla Dungeons & Dragons 5th Edition (2014), Inspiration operates identically to having Advantage. Advantage lets you roll a second 20-sided die on an ability check, a saving throw, or an attack roll—and you keep the higher roll. Advantage is simple, elegant, and fast moving. It does, however, rely on very specific skills, feats, or your dungeon master's discretion in order to create.
Inspiration, on the other hand, lets you keep that second d20 up your sleeve for use on any ability check, saving throw, or attack roll the player wants, regardless of any external conditions being met (to include your DM's permission). But Inspiration was originally meant as a reward for roleplaying your character's personality traits, flaws, and bonds. As excellent as Inspiration is, it was dependent on whether you were getting into character. If you're feeling gun shy about some trait, flaw, or bond you conjured during your Session 0, or the person your character was becoming wasn't in keeping with the character you started as, then a DM might withhold Inspiration, regardless of how well you may be playing this out-of-character character. Not ideal.
Or perhaps you have a DM like me, that literally puts you through Hell (or Avernus, rather) and thinks that every day you wake up in that fiery, war-torn realm, you wake up in the un-morning with Inspiration. I meant it as a heroic boon for my players that were fighting the good fight down in a plane of existence that is doing everything it can to turn your game into an evil campaign.
While it's unclear if vanilla 5e's traditional method of dealing out Inspiration is being deleted, D&D 2024 is introducing Heroic Inspiration. You will no longer roll with Advantage; you won't roll two 20-sided dice and choose the higher roll. Heroic Inspiration is a reroll; you roll that d20, see the result, decide you want a higher roll, then reroll that d20. You're stuck with the new result, whether higher or lower.
Also, Heroic Inspiration isn't just for d20's. You can reroll any roll one die. Whether it's the low die on that 2d4+2 hit points you got from that healing potion, the wackadoodle d100 result on a Wild Magic table, or the d2 flip of a coin determining your fate with an old god. Certain players are already making sure they've got Heroic Inspiration going into their level up so they get a second shot at that low roll for more hit points.
Designer Jeremy Crawford doesn't discuss how Heroic Inspiration is handed out in D&D 2024. Is it still dependent on those traits, bonds, and flaws from vanilla 5e? Is it something more akin to my haphazard method of doling out Inspiration like taking your morning vitamins? We shall see.
