"Bonds" section in the Player's Handbook with red dice and a classy filter.
I like the idea of bonds that specifically tie adventurers together, not just the generic options that are provided in the Player’s Handbook. This allows the characters to act on something beyond basic self-preservation and self-enrichment and also gives the DM a reason to award inspiration.
Process is simple, each player rolls a d6 and the resulting stat is boosted on their character. I generally use the same order that is on the character sheet (1 increases strength, 2 increases dexterity, et cetera). The players then have to make up a short, heroic back-story about something that their characters did together to warrant such stat increases. The players should also indicate whether their characters are friends, enemies, or somewhere in between.
For example, Tim and Eric decide that their characters were friends in the village. Tim rolls a 4, which increases his character’s intelligence, while Eric rolls a 1 which increases his character’s strength. They make up a quick story where Tim’s character, Ario, figured out a plan to scare off an orc patrol, and Eric’s character, Lara, led the charge to successfully scare them away. The bond should be written down on the character sheet.
Take notes, it’s a wealth of hooks
The DM should listen to the player's story and then ask pointed, leading questions. What clan were the orcs from? Who got hurt during the plan's execution, and now resents Ario and Lara? The DM should make quick notes for later story hooks. Those orcs are clearly coming back...
Living together
The presupposition in doing a bonding stat is that the characters already know each other. They either grew up together in the same village, or at least have been in contact for some time. Most of the time, this can easily be worked into the story. Is the game starting in a prison? Perhaps the characters have been in lockup together for several weeks, what explains those stat increases? Maybe Tim’s intelligence and Eric’s strength was used to scare of a gang of rival prisoners as opposed to orcs.
An option for the standard array?
A point and a half of ability score is what separate the two standard methods of character generation: the standard array produces 72 ability points and 4d6 drop the lowest on average produces 73.5. Adding an additional “bond stat” for the standard array brings the methods closer together and a bit of randomness to an otherwise static character.