Billy Needs an Organ

Since in Real Life organ donation is still an opt-in process (you have to sign up to become an eligible donor) in most countries, lots more people need transplants than there are organs. Rule of Drama means that this pops up quite a bit in medical and crime shows. Often a Knight Templar Parent will do whatever it takes to get the Littlest Cancer Patient the organ he needs, especially if the person ahead of him is a Rich Bitch who ruined her body by hard living. Less often the Rich Bitch learns An Aesop about how precious life is and to open her heart to others (especially if said heart was what got transplanted). Even more rare are incidents in which the organ doesn't take (unless the series is a medical drama); in horror movies, however, the biggest side effect is an increased risk of Grand Theft Me.

Related tropes: Organlegging, Cloning Blues (when the clones end up being a source of new organs).

Sister trope to Healthcare Motivation.

Examples of Billy Needs an Organ include:

Anime & Manga

Angel Beats! an anime set in purgatory, more or less, reveals in episode nine that just moments before Otonashi died he signed a donor card, in the hopes his death wouldn't be in vain.
- It's shown in episode 13 that indeed, it wasn't in vain as Kanade has his heart.
Adorea of Franken Fran has this in her backstory: she and her boyfriend had life-threatening illnesses, and they each promised they would give their organs to the other. The boyfriend died first, so Fran's father donated the organs to Adorea before working his magic on the boyfriend to revive him; the callous boyfriend demanded the organs back, turning Adorea into the...uh..."person" that she is now.
Purely hypothetical in Ichigo Mashimaro: Miu is claiming that her friendship with Chika is closer than that of Ana and Matsuri. One test is what one friend would do if the other needed the first's organs. Matsuri and Ana would definitely do so, even though Matsuri asks what would happen to her if she were the donor (Matsuri seems to forget for a moment that it is hypothetical and starts crying); Chika is more ambivalent, but comes to the conclusion that she "probably" would, provided that she herself would live. Miu claims "there's no way I'm giving you any of my organs, though."
One of the murderers in The Kindaichi Case Files has trying to find a matching kidney for his sick daughter in his backstory—he got involved in the Organlegging business, and the man he murdered found out and was blackmailing him over this.
A running theme in Ray and the animated series based on it. Ray herself was raised on a farm designed to provide black market organs, this was how she lost her eyes, only to have them replaced by X-Ray eyes.
In Pet Shop of Horrors, the dictator of a country is hiding out in America and is in need of an organ transplant. The doctors decide to use the heart of a baboon to do the job, only for said baboon to be kidnapped. Cue a mad rush to find the thing, for fear of sparking a messy political situation. And then it turns out that the kidnapper is a man whose daughter also needs a heart transplant. The dictator winds up being sniped by someone from his country and they end up giving the baboon's heart to the girl after all. Oh, and there's a subplot where D is apparently suffering from some disease and his "sister" is sent by their father to give her dear brother any of her organs or body parts to cure him. She herself is actually a baboon, and is killed by the shop's pets when D refuses to accept her offer and she attacks him. Needless to say, it was a creepy story.
In Tokyo Babylon, an Ill Boy named Yuya had an unidentified kidney disorder, which had already killed his twin older sister Maya. Even though his condition kept worsening to the point of emergency surgery he was still on the waiting list, so Subaru eventually decides to donate his own kidney to Yuya. He arrives just after Yuya's taken to emergency surgery again, and Yuya's unnamed mother flips out and attacks him. Seishiro blocks the knife, and Subaru is too upset to do anything. The plotline is never resolved.

Film

The Clint Eastwood film Blood Work has a detective investigating the murder of the woman whose heart he got.
In The Eye, a woman receives an eye transplant that helps her see into the supernatural.
The Anvilicious John Q. has the title character taking a hospital's emergency room hostage to get his son a heart transplant he can't afford to pay for and his insurance refuses to cover. Further, when there's no suitable donor heart available, he starts planning to kill himself to provide one. He's stopped seconds before he pulls the trigger, when one suddenly becomes available.
Seven Pounds is about The Atoner looking for worthy recipients of his organs before he commits suicide.
This happens in Awake when Clay's mother kills herself in the hospital waiting room to assure he gets a healthy heart.
Scifi variant in Star Wars: When R2D2 returns from the attack on the Death Star with some structural damage, C3PO offers to donate any of his circuits or gears that might be needed to repair his companion. Awwww....
In Return to Me, the female lead's recent heart transplant provides some major plot when she finds out the guy she's dating is the donor's widower.

Literature

Never Let Me Go starts off as a slightly creepy story about some kids in a boarding school. About a quarter of the way through we find out that the kids are clones being raised to provide organs. By the end we find out that most clones are indoctrinated so much that they keep giving organs until they end up comatose organ mines.
This turns out to be the basic underlying scheme in Strange Affair by Peter Robinson.

Live Action Television

One CSI episode centered around a family with an ill son whose younger sister was conceived for the express purpose of finding a match for a bone marrow transplant.
- The Jodi Picoult novel My Sister's Keeper features the exact same situation.
- There was also a black market bone/tissue market discovered in another episode, when strange objects used to replace the harvested items were found inside corpses.
CSI: NY 'Live And Let Die' is basically this trope, with the variant that it was the doctor's wife needing the liver. He orchestrates a medical helecoptor hijacking and kills several people in the process. Mac is naturally not amused, especially recalling his own pain and loss and talks on the fact that the guy will likely be in prison while his wife dies.
Grey's Anatomy, many times. The most noteworthy is Denny Dukett, who fell in love with Izzy and was about to get a donor heart when the donor died, meaning that there was only one heart left at that hospital, and it was going to go to another person. Izzy cut his LVAD wire, purposefully worsening his health to bump him up the list. He survived and got the transplant, but died later of a stroke.
Earth: Final Conflict: Liam Kincaid anonymously donates blood to Ronald Sandoval. Notable because the doctor said it had to be from a first-degree relative (parent or child) to work. Furthermore, Sandoval considers Liam his enemy while Liam's feelings for his human father are a little more complicated.
House. Oh so many times. There's an episode where a father kills himself to give one of his organs to his son, another where a patient has to find a kidney on the black market, and another where they have a brain-dead donor who'd be perfect for a man with an ailing heart—they just have to figure out what was wrong with her so that the donor board will sign off on it. Then there was the time they had a donated pair of lungs that were going to be implanted but couldn't be because they had some kind of illness, so House was called in to cure them when they were nothing more than a pair of lungs in a box.
There's an episode of Law and Order where the villain is giving away his organs to people he feels deserve it, and killing them (in such a way that the organ can be transplanted into someone else) if they stop whatever project or behavior he feels made them deserve it. In the end they get him to confess in exchange for allowing him to continue donating his organs while in prison.
- Another example features a millionaire philanthropist who is so desperate to acquire a kidney for his dying daughter (with the added complication that, due to a rare condition, she will reject any organs provided from any donor who is not a specific match to her blood type) that he bribes a doctor to kidnap some guy, steal his kidney then dump him in a park.
  - This episode may have contributed to the persistence of the "kidney theft" urban legend.
NYPD Blue: Bobby Simone needed a heart but was far down the list, so they asked all the cops in NYC going out on patrol that day to make sure they sign up for organ donation and stipulate it goes to a cop, Just In Case. The one who ended up dying didn't sign up, and they had to convince his widow to allow them to take his heart. Bobby died anyway.
One of many Court Cases Of The Week on Picket Fences concerned an elderly man with Alzheimer's, who wanted the legal right to commit suicide under controlled conditions in the hospital so his dying son could receive his heart. A very moving episode, in that the plaintiff was coherent at certain times of day (the "sundowning" phenomenon), and plainly incompetent at others. The son eventually gets the heart of a homeless man who'd frozen to death instead ... only to wind up shooting his dad to spare his dignity after finding him stark naked astride a child's rocking horse. Cue the next Court Case Of The Week.
In the Scrubs episode "My Lunch", a recurring character commits suicide and three other patients receive organs from her. However it turns out that she didn't die of a drug overdoes, but of rabies and now the other patients have it too. All three die, and while two of them would never have survived long enough to find another donor anyway, one could easily have held on long enough if they hadn't been infected.
- Truth in Television: This episode was based on a real case.
On Psych, someone kills a number of people on the regional liver transplant list, in an apparent attempt to move their own name to the top.
- Also a You Fail Your Medical Boards Forever, as the woman in need of the transplant has an identical twin, who should've been able to voluntarily donate a liver lobe to her sister without recourse to the donor list.
In one episode of Babylon 5 Londo ends up needing a blood transfusion, but has a very rare blood type. He is saved by a transfusion from his wife Timov (Who stated that blood type was the only thing they had in common). While she gets Dr Franklin to promise to not say where the blood came from, the novels about Londo's reign as Emperor state that he eventually figured it out.
On Leverage, The Cross My Heart Job features a rich businessman who attempted to steal a heart that was intended to be transplanted for a teenage boy. Unfortunately for him and fortunately for the boy, Nate and his team find out about this and are able to recover the heart. It is interesting due to the limited amount of time that the team has to work with.
Forever Knight did this in season 3. Nick and co are investigating a black market organ ring. Natalie is scheduled for a knee surgery, and has the misfortune to have as her doctor who's connected to it somehow and who's the mother of a girl who needs a heart transplant. She nearly ends up an unwilling donor, but Nick gets there just in time.

Video Games

Jade Empire has a sidequest involving a young girl whose liver is dying, and her father has come up with a way to replace an organ, but he needs a new or preserved liver. Your options are to convince him to use his own liver or to convince a ghost to atone for his sins by giving you his preserved liver.

Western Animation

Batman and Mister Freeze Sub Zero, where Mr Freeze's wife Nora will die without an organ transplant. Notably, they get through the entire movie without ever specifying which organ it is that she needs; it's implied that she needs several. It turns out she just needs a blood transfusion, but from a very compatible donor... who happens to be Barbara Gordon aka Batgirl. By the end of the movie, Nora is saved.
Dr. Venture finds himself in need of two kidneys after he wakes up in a tubful of ice and a kidney-shaped set of stitches in his back. (He ends up taking one from each of his sons, who have been cloned multiple times because each keeps dying in some stupid fashion.

Real Life

When Alanzo Mourning needed a new kidney, he got countless offers from his fans to get a new kidney; Mourning told them to register their names with the Red Cross instead, and is currently lobbying for legislation to make it easier for others to get new kidneys.
See this story
One of those sidebar stories in the Reader's Digest tells the tale of a woman who received a heart transplant and (unknowingly) struck up a romance with the grieving widower, wondering why he'd always caress her chest where her heart lay.