There's enough ways to get advantage in D&D without inspiration that it's no wonder players ignore it–and why we explore variants. I want inspiration to promote shared storytelling at my table. I borrow a bit from Fate and let players spend Fate Points to add aspects to NPCs and locations. Want to make that duchess paranoid so she's more receptive to the party's tale of an Aboleth controlling her children? Go for it. She'll be a more memorable character than whatever I planned because my players are invested.

I offer fate points to players when their characters fail a d20 roll if they accept a story-related complication. The player doesn't have to accept, but I try to make the complication increase excitement at the table. If I can't think of a complication that fits, the game just keeps moving.