Clue (or Cluedo, for those of you outside my continent): You've been invited to a dinner party thrown by wealthy mogul Mr. Boddy at his stately Tudor Mansion. When Mr. Boddy winds up dead, you have to figure out whodunnit, with what weapon, and where. The problem is the alluring female suspects, Miss Scarlet, Mrs. White, Mrs. Peacock, and Dr. Orchid, all seem to be doing whatever they can to try to distract you from your investigation.
Guess Who: I've had this idea for a while of how the classic guessing game would lend itself to narrative fiction. You're a sketch artist for the police, routinely creating illustrations of suspects by asking witnesses questions like "Did they have a hat?" "Do they have a mustache?" "Do they have big ears?" You're at a convention when a crime is committed and you have to use your talent to find the culprit. But as in the previous scenario, the female witnesses/suspects, Anita, Anne, Claire, Maria, and Susan all seem to be doing whatever they can to keep you distracted.
Bang!: A spaghetti-themed card game much loved by many, including some of my closest friends. Plenty of unique characters and character art. Everyone is either the sheriff, a deputy, an outlaw, or a renegade. The sheriff and the deputy are trying to kill all the outlaws. Everyone knows who the sheriff is but no one knows who the deputy is. The outlaws are trying to kill the sheriff and the deputy. The renegade is trying to kill all the outlaws, the sheriff, and the deputy. And whoever you are in that mess, this town's full of femme fatales out to make it even more complicated for you.
Barbie: Not a board or card game per se, but the famous line of toys/dolls is rife with potential for this kind of story.
My taste in tabletop games may be a little basic so, as always, feel free to add to the list.