Persuasion grenades are a unique type of grenades with only a single application: making settlers more open to suggestions. While they can be used at any time, they are effective only during quests to take over Commonwealth settlements for the Nuka-World raider gangs. Tossing one at the feet of the settlement's leader will result in an explosion of faint whitish gas and an on-screen notification that the settlers in question are now easier to persuade. Doing so effectively reduces the difficulty of the speech checks required to persuade the settlers to leave their land peacefully, provided this path of action was chosen prior to starting the mission.
Persuasion grenades can be acquired from Lizzie Wyath during the quest Home Sweet Home. An optional target will have the Sole Survivor ask the raider gangs for support in taking over the Commonwealth. Talking to the Operators will eventually yield a single grenade along with the persuasion grenade recipe to craft more.
One located in the Nuka-Town market inside of a shopping basket on a table near the arms dealer Aaron Corbett. This is a reference to an entry on Lizzie Wyath's terminal that mentions an attempt to test her prototype persuasion cocktail on Aaron.