This is true not just of adventures, but of all games... but what I love in adventures is when they transport me to a place that is new and inspiring, to meet characters that are charming, funny and touching. It is something that many of the classics manage still - I played Monkey Island for the first time a couple years ago and loved it - while more recent offerings just don't seem as inspired.
I want something honest that doesn't try to be hip or clever.
The response you want from a game preview is not: "Hey, that could be alright" but "Wow! I always wanted to be that character or visit that place (even if I never knew it till now)! I must have it."
Games that do this (and yes I'm including non-adventures): Assassin's Creed, Brutal Legend, Okami, Shadow of the Colossus, Monkey Island, Phoenix Wright, Grim Fandango, Gabriel Knight. These are characters and places worth inhabiting.
I refer everyone to Tim Schafer's
Greater Mech-suit Theory.
"Finally the game that satisfies the player's desire to be a mentally insane ostrich."