Cheers for starting this thread!
My old campaign was based on a homebrew setting using this premise: every monster has a story, and every story tells its downfall. Most of my monsters were elites or solos, either home-brewed or reskinned...minions were their followers and standards their lieutenants.
For example during 4th and 5th level, the PCs investigated the striga, a type of monstrous shapeshifting vampire (my version a shameless ripoff from The Witcher). Basically they learned someone in the royal family was cursed as the striga but not who, and the king had contracted them to kill the striga, while the queen wanted a commoner framed...it was a political minefield.
Three things freaked the PCs out about the striga and made her stand out in their minds as not just another monster...
1. Build-up: They saw the effects of the striga's precarious on the frightened townsfolk, the mad gravedigger, the castle's tightened security, and the mirrors...oh the mirrors. Any time the striga was in monster form it burned a blurry image of it's human form on mirrors and other highly reflective surfaces. Man I creeped them out with that, even got a little scream from one of the players when the wind slammed a door shut. >)
2. Ambiguity: Their research turned up several possible origins of the striga, but the last written account of one was 115 years ago, and the attacks from 7 years ago were supersede by agents of the crown. Was the striga aware of it's nature in human form? Was there a way to break the curse without killing the striga? Was the striga being controlled for political killings or was it killing based on primal urges? Was it the princess who was locked away by a convent? Or the queen? Was it traded like a role, so there was more than one striga? Was its curse sent by the gods, and what would be the ramifications of challenging divine will? And, since it's poison bite could drive others mad, was it possible the striga herself was bitten by a far older monster?
3. In the Shadows: Their first confrontation with the striga came after an intense bit of roleplaying and a skill challenge. Realizing who the next target would be, the PCs went to the target's house ushering the family into the impenetrable wine cellar while they readied themselves for battle. The striga started killing horses to lure out the PCs but when that didn't work she unnerved the PCs by dragging her nails along the windows and scrambling across the roof; the mauled dying guard at the doorstep really freaked them out. With surprise, the striga broke through a window, grabbing a squishy PC then dragging him out the window into the night screaming. Once the battle began the PCs unloaded a world of hurt before the striga fled (six 4th level PCs vs an 8th level solo skirmisher/lurker), leaving one PC near death and infected. The striga got owned...but even after the fight the monster remained scary.