Picky People Eater

""A census taker once tried to test me. I ate his liver, with some fava beans and A Nice Chianti.""

- Dr. Hannibal Lecter, The Silence of the Lambs

Humans, like most living things, are composed of many different organs and tissue types. Some people-eating monsters out there will not find all of those parts appetizing enough to eat.

A monster can be made more horrifying if instead of simply eating people, it eats only certain parts of people. The vampire is the best known example of this type of monster with its interest in blood. But for every organ or tissue type composing the human body, there will be some critter that finds it a delicacy.

The horror quotient can really be ramped up if the part the monster is interested in can be extracted without immediately killing the victim. The victim can then suffer a slow, screaming death in front of their companions and the audience as the monster painfully extracts its chosen food. Also a partially consumed corpse can be stumbled over announcing to the audience and the characters that something is rotting in Denmark.

I'm a Humanitarian is the supertrope if the feeder is also a human (although, technically, this also works for other other species that eat their own kind). If he's not then see To Serve Man. Brain Food is a subtrope.

Anime and Manga

 * The mushi in Mushishi were often fond of eating strange things. People could have their eyes consumed, their memories consumed, their dreams consumed, their "life force" consumed, etc. One particularly strange example involved a mushi that could take a recently dead human corpse, eat the "lifespan" of that corpse and revert them into an embryonic form that could be implanted into a pregnant woman. (Of course, most of these examples aren't nearly as squicky as the usual examples of this trope, and some people were even able to benefit, at least temporarily, from the mushi taking over and/or consuming some part of their body.)
 * Natsume from the Mermaid Saga Came Back Wrong after being brought back from the dead using a mermaid's liver. As a result, she now exclusively eats livers, be it those of animals or humans.
 * The Youma and Awakened Beings from Claymore prefer the entrails.

Comic Books

 * The Corinthian in The Sandman comics likes to eat the eyes of his victims—with the little mouths that he has in place of his own eyes. In doing so, he is able to see the last thing they saw before they died.
 * In Lucifer there was a monster that fed on humans. A demon offers it a human she has befriended..
 * Batman foe Cornelius Stirk operates under the delusion that he requires the nutrients and hormones from peoples' hearts in order to stay alive, and these are best prepared with norepinephrine by inducing fear in the victim prior to death. In other words, he terrifies people, then kills them and eats their hearts. You can probably guess what era of comics decided they needed a Darker and Edgier People Eating version of Scarecrow.
 * Emplate of the X-Titles feeds on mutant bone marrow.
 * The Libra Killer in Alan Moore's Top Ten fed on human pineal glands.
 * In Grendel, Tujiro was an Asian vampire who consumed eyeballs as well as blood.

Film

 * Island of Terror had the bone-eating Silicates.
 * Deep Rising had a creature interested only in human bodily fluids.
 * The modus aperandi of The Creeper from Jeepers Creepers was that he'd frighten his victims, releasing pheromones which he'd sniff to detect whether or not they had the... pieces he wanted (or needed to replace lost bits). He'd proceed to remove body parts from his chosen victims, consuming the smaller parts like eyes. Whether or not the victim was still alive at the time of "extraction" didn't seem to matter to The Creeper.
 * The 1997 movie The Relic featured a creature that could only survive by consuming a specific protein found in a certain Amazon plant and human hypothalami.
 * As above, Hannibal Lecter in his series of books and movies. He's even picky as to which people he'll eat. Also, due to Lecter's medical skills... they didn't have to die before he, um, went to town.
 * Imhotep in TheMummy is probably one of the pickiest people eaters around. He took body parts only from the people who had basically stolen his.
 * The "tooth fairies" from Hellboy II: The Golden Army eat the whole person, but go for the teeth first.
 * Braaaaaainnnnssss...

Literature

 * Orson Scott Card's short story Kingsmeat. Yup, it's exactly what you think it is. The alien king and queen would kill a person and only use a small part for their meals, so the 'shepherd' convinces them to let him merely remove what they want and let the people keep living. When the people are liberated, the people reward the shepherd with many gifts, and also remove those body parts he doesn't strictly need to survive (though they leave him his eyes).
 * Wicked!: Paul Jennings & Morris Gleitzman's children's book (no connection featured the Slobberers, slug-like nasties that sucked the bones right out of their victims' bodies. While still alive.
 * Ella Enchanted contains a scene in which the heroine is captured by a family of ogres, who argue over how to divide up her parts. (The neck is the best part, apparently.)
 * Ray Bradbury's short story Skeleton.
 * Some Cthulhu Mythos monstrosities have weird dietary preferences for blood, brains, marrow, etc.
 * The Deltora Quest children's books feature the Granous, large furry humanoids who prefer fingers and toes, eaten one by one as they play sadistic riddle games with their victims.
 * The skyskirr from Secret of the Sixth Magic by Lyndon Hardy subsist on bone marrow, which they can somehow drain from living creatures' limbs without necessarily killing them.
 * Santa Steps Out: The Tooth Fairy lived on teeth and bones although she had a few other uses for the rest of the body. You really don't want to know where that dime under your pillow came from.
 * The dimes upon your eyes?
 * In The Amazing Maurice and His Educated Rodents the sentient rats don't object to eating each other, but they always spit out the "green wobbly bit".
 * The Warhammer 40,000 Night Lords series has Uzas, who shows a great fondness for eating geneseed, preferably from the still living flesh of loyalist marines. One could take it solely as a sign of his status as the resident Ax Crazy, but even the hero Talos makes mention of the practice, and later eats the heart of a Blood Angel he killed in an of revenge.
 * A Bastion Wars novel once shows a rebel insurgency of Chaos cultists trying to liberate their planet, and they had a tendency to cut out and eat the livers of their Imperial counterparts.

Live Action TV

 * The X-Files enjoys this trope.
 * Eugene Victor Tooms fed on human livers.
 * The "fat-sucking vampire" that fed on overweight women. There's a business model right there.
 * Also doubles as a case of Did Not Do the Research. All he needed was a spoon and a small-town diner that cooks with lard. Just wait until everyone goes home, go behind the diner and pop open a 55-gallon drum of used lard, and chow down. More importantly, you're not going to get a prison sentence for being caught eating used lard.
 * A monster that replenished itself by eating tumors.
 * Buffy the Vampire Slayer
 * Gnarls have a thing for skin.
 * There are other species mentioned as having this habit, like Small Bone Eaters and Large Bone Eaters. (Note that those names refer to the size of the bones they eat, not the creatures themselves.) For that matter, vampires themselves technically count, since they're only interested in the blood.
 * Angel: A demon from ate only the hearts of heroes. This gave Angel a bout of low self-esteem when it didn't want his, though a couple people pointed out that maybe it just didn't want to eat a heart dead for two hundred years.
 * Sliders: Among the Kromaggs human eyes are a delicacy. The series' first recurring Big Bad, Colonel Rickman, suffered from a degenerative nerve disease that required him to regularly steal brain fluid and inject it into himself to survive. His victims were left in comas.
 * Farscape: There was the girl who ate bones. And Scorpy's girlfriend who liked to eat eyes. Especially blue ones.
 * The Fringe episode "Midnight" has a woman who's been turned by mad scientists into a cerebrospinal fluid sucking vampire.
 * Criminal Minds: Eddie Mays from "Blood Hungry" ate a body part believed to contain the soul. Of course, so many religions believed the soul existed in a body part and they were all different, so Eddie just wound harvesting a different one from each victim (liver, stomach and heart).

Music

 * The eponymous monster of Sheb Wooley's "The Purple People Eater" only eats.

Mythology

 * The kelpie in myths of the British isles would take the form of a horse and lure children on to its back, where it would eat their insides—except for their kidneys or livers, which would float to the surface.
 * Oriental mythology is full of these.
 * The kappa of Japanese mythology would pull children (and sometimes adults) underwater and suck their intestines (or blood, or liver, or ki/lifeforce—the mythology is inconsistent on just what they ate) out through their anus (this part, however, is pretty consistent). You could only protect yourself from this fate by carving your name on an even tastier treat and tossing it into the water. The one food they preferred over human innards? Cucumbers.
 * The kumiho of Korean mythology (a form of nine-tailed shape-shifting fox spirit with a serious mean streak in later myths) is said in various myths to eat people, but only one particular organ. In some, it's the heart, while in others, it's the liver, as that's where your Life Energy is supposed to reside.
 * The Aztec ahuizotl only ate eyes, toenails, and fingernails.
 * It is common among cannibalistic peoples not only to be picky about what parts to eat but about which people to eat. As eating is often held to absorb someones moral characteristics as well as their physical ones, they go out of their way to be sure to eat a Worthy Opponent . In other words if you ever get eaten just consider it a compliment!

Tabletop Games

 * Dungeons & Dragons is stuffed full of these such as the bonedrinkers which, funnily enough, drink their enemies' bones. Inverted with the Chuul, which eats everything except for the brain, which is toxic to them. Mind Flayers, who play this straight, keep them as pets to throw brainless corpses to.
 * In Vampire: The Masquerade, this is the clan Flaw of the Ventrue. Being of such refined taste and upbringing, they can only partake of blood that comes from a certain kind of subject. This varies from Ventrue to Ventrue, with such "tastes" including virgins, gay men, priests, etc.
 * In the successor game, Vampire: The Requiem, this quality leaves the Ventrue (in favor of a whole different Flaw), but shows up in some of the bloodlines, such as the Morbus (can only gain sustenance from those with a disease) and the Anvari (can only drink from those who are high on narcotics).

Video Games

 * In Xenosaga, Realian nerve tissue contains compounds that act as an addictive drug.

Western Animation

 * An episode of Stroker and Hoop had the eponymous characters' organs stolen by a "friendly" cannibalistic cult that only ate the non-vital organs of its victims.