Heroic Inspiration may be spent in the following ways, in addition to the normal effect of preemptively granting oneself Advantage on a roll they are about to make:
You can spend Heroic Inspiration after a die roll to simply reroll it, rather than having to spend it before and not need it.
It is worth noting that Heroic Inspiration has a one per round limit of expenditure, so you cannot spend Heroic Inspiration gain Advantage, fail, and then have another player grant you their Heroic Inspiration to attempt an additional reroll.
A player can spend their Heroic Inspiration to "remember" their character purchased a mundane item (up to and including basic Potions of Healing) at some point in the past. You may only do so if you have the requisite amount of gold, however, which you must deduct to reflect the purchase. This rule assumes that Adventurers tend to carry all sorts of things in their packs, sometimes they carry so many things that they lose track of them, only to remember them at just the right moment.
A player can spend their Heroic Inspiration to declare a feature exists in a scene if they'd rather not ask. This speaks to the frequently asked questions like "Does this room have a chandelier?" or other object the players wants to interact with to do something unusual. Normally you would ask and there would be some manner of negotiation, and we would proceed. This rule simply allows you to state the thing you want is present (within reason) at the cost of your Heroic Inspiration.
This can also be used to set up reasonable flashbacks where you might have prepared for something in advance.
A player can spend their Heroic Inspiration to describe a contact who can help with the current situation. Quickly define a few details about this NPC, their past relationship to the player, and how the player believes they can help. We then make a Charisma Test, the Difficulty of which is determined by how many specific details the player wishes to define about this contact.
Depending on the situation, the check can be made with Advantage (such as looking for a criminal in a bad part of town) or Disadvantage (such as looking for someone with a highly specific set of skills or is unlikely to be available to the party, such as a king).
Depending on the roll, and the nature of the NPC, the contact might be immediately helpful or the relationship between the two might have soured, making them less likely to be immediately helpful without pay or some other promise of recompense.
On a particularly low roll, the NPC might even enter the story as an antagonist and, in the case of particularly outlandish requests, they might be declared completely unavailable. In either of these cases, the Heroic Inspiration spent is refunded.
It is worth noting that neither "Wouldn't it be cool if..." nor "I know someone for that..." can be used to completely solve a mystery or bypass an adventure. Their intent is to help the players accomplish their goals by allowing them to have slightly more authorship over the world. Any attempt to "summon the murderer" or "Materialize a lethal pit trap under the bad guy" will not be accepted.