Text version:
Inspiration
When a friendly creature within 30 feet that is aware of you fails an attack roll, an ability check or a saving throw, you can Inspire that creature to make it reroll the die.
Limitations. (a) You can only Inspire another creature once per long rest; (b) If you give or receive Inspiration, you can't give or receive Inspiration again until the end of the current round.
Flavour is a welcome addition. When inspiring another creature, you are invited (but not required) to describe how you are doing it: for example, you may hype them up, warn them of an incoming danger, or throw a passing comment that may help them achieve success in a task.
No beggars allowed. Inspiration should feel genuine and come from the heart: this mechanic is meant to incentivize bonding and roleplay, not min-maxing. At the DM's discretion, you lose the ability to give and receive Inspiration until you finish a long rest if you ever explicitly refer to this mechanic, such as asking others to give you inspiration, or making a throwaway comment about it (for example: "Could you please give me inspiration on this roll?", "It would be cool if somebody could help me out on this roll", "Man, I feel really inspired right now", "Don't waste your Precision Attack on this roll, I'll just inspire you if you fail").
This mechanic is meant to replace the original inspiration. Instead of the DM awarding Inspiration to a player, it's the players themselves who help each other during hard times.
Whops, I forgot to hit send on this one, so I'm writing this comment a tad bit late.
In DnD, helping others is limited to the Help action, which nobody takes because it's such a waste of an action. I wanted something to spice things up, something that made players feel like they were a tight group helping each other out instead of a band of mercenaries covering different grounds.
To do so, I looked at XCrawl's Mojo system, which this variant Inspiration is heavily based on. As with the original, it's designed as an incentive for bonding and roleplay, hence why you can only spend Inspiration on other players'rolls.
To simplify things, I did away with earning additional Mojo/Inspiration points and losing Mojo/Inspiration on a nat1: instead, the players have a single Inspiration point every day, that they can spend to help other players in a clutch.
The "no beggars" limitation is also lifted wholesale from XCrawl. I really like it because it awards players for staying alert during their fellow party members's turn.