Older Than Radio: Fantine in Victor Hugo's Les Misérables. She subverts a few characteristics of this trope, popular in that era and genre, because her illness is named, her wizened, aged appearance is described in detail, and when she dies, she does not die from the disease itself. On the other hand, she manages to soliloquize paragraphs on what she's going to do when she gets better (despite the fact that the narration says she coughs near-constantly) and win the heart of all around her.
Fantine in the musical of Les Misérables is ailing and wasting away, and also sings a plaintive song about the winter wind crying and the night encroaching, and of how she would love to see her little girl one more time before she dies. And then she dies. And her death acts as a motivator for the rest of Valjean's actions for the next, oh, ten years or so.
Elizabeth "Beth" March from Little Women, who ar first was just a little delicate, but later contracted scarlet fever and the side-effects gravely weakened her. She later became the March family's dead little sister.
Many of the female characters created by Edgar Allen Poe fit this type. Notable examples include the eponymous heroines of his short stories "Ligeia" and "Berenice", and his famous poem "Annabel Lee". They always die.
Michelle in Robin Cook's medical thriller Fever, suffering from leukemia and further weakened by chemotherapy.
Queen Ehlana, in The Elenium trilogy by David Eddings, is a variation on the trope. Under normal circumstances, she's perfectly healthy and energetic, but as it gets explained to her personal champion, she's been getting progressively more sick since her coronation, and her sickness isn't something that anyone's ever seen — her symptoms contradict each other. They eventually work out that she's been poisoned, and the poison she was given has no known cure. Except one.
Two, actually. The problem is that the first candidate had already been destroyed by a healer who knew of its healing power but not that you could use it without destroying it.
Diggory's mother Mabel Ketterley-Kirke in The Chronicles of Narnia: The Magician's Nephew is bedridden and dying of an unnamed disease. In fact, she and Diggory live with Uncle Andrew and Aunt Lettie because they're taking care of her while Mr. Kirke has to work in India. With a little help of Aslan and a magical apple he gives to Diggory at the end of his and Polly's adventures, Mabel ultimately gets better.
The aforementioned Kate in My Sister's Keeper, with leukaemia. Arguably, My Sister's Keeper is a deconstruction of this type of story, showing how Kate's mother's efforts to save her daughter take a considerable toll on the whole family.
Kate is actually so ill that she actually does want to die, thus she asks her sister/prospect donor Anna for help so Anna can be released from being her forced donor and Kate can die in peace.
This trope is also seen in two of Jodi Picoult's other books, Handle with Care (Willow has ostogenesis imperfecta, or brittle bone disease) and Change of Heart (Claire Nealon needs a heart transplant). Like My Sister's Keeper, the mothers of these children are willing to do anything for their child (in the former, the mother sues her best friend and OBGYN for not telling her about a disabled child, that she presumably would have aborted, and in the latter the mother refuses a transplant from the murderer who killed her husband).
The book María Jesús: Un milagro de amor ("María Jesús, a miracle of love") by Chilean author Ana María Figueroa is the biography of an Ill Girl named María Jesús, gathering all kinds of testimonials about how she coped with the leukemia that ultimately killed her.
Laura and Eileen from Laura and the Silver Wolf both have leukemia.
Alice in A. Sapkowski's Złote Popołudnie ("Golden Afternoon"), a retelling of "Alice in Wonderland" from Cheshire Cat's POV. In this version Alice's visit in Wonderland is in fact her DyingDream after drinking laudanum instead of her cold medication. She does get better.
Gabrielle Gabrielson in The Story of Gabrielle by her mother Catherine. Published as a Real Life story, although there's no background info to verify it. Gabrielle, the brilliant and Spoiled Sweet little daughter of a well-to-do New England couple, begins complaining of stomach pain. She informs her mother that the pain is incurable and will eventually kill her. She's right. While most of her doctors think she has hepatitis and a few dismiss her as play-acting, a tiny "shadow" on the x-ray troubles the youngest doctor enough to call for an exploratory. He finds a gigantic, malignant cancer at the base of her spine. Catherine records Gaby's last weeks, her strange speeches and experiences as the cancer invades her brain, and her quiet death. All of this takes place in a hospital, which seems unlikely until you remember this was the 1940s and people could actually afford this level of health care.
Cadpig from The Hundred and One Dalmatians by Dodie Smith, who was born nearly dead and revived, and is significantly weaker than her siblings throughout the book, requiring special provisions to be made for the journey back to London.
Sister Edith from Selma Lagerlöf's Thy Soul Shall Bear Witness!, a Salvation Army worker who caught an Incurable Cough of Death while at work and has a single wish to be fulfilled: to speak with a man named David Holm, whom she has tried and failed to redeem. Little does she know that David has died and his soul has been forcibly made into the rider of the Death Cart, so he can see how he has ruined other people's lives. As such, David is the one who fetches Edith's soul — thus sorta fulfilling her wish.
Evangeline "Eva" St.Clare is a particularly melodramatic example in Uncle Tom's Cabin. Before dying of her tuberculosis, she convinces the sympathetic characters to change their ways and resist the evils of slavery.
In Jack and Jill, Janet "Jill" Pecq is severely injured in an accident and almost is confined to a wheelchair. The book is about her and the other wounded kid (her boyfriend Jack)'s shared path to recovery as well as their Coming of Age Story.
In the backstory, Mrs. Minot's childhood friend Lucy Snow spent 25 years bedridden after a similar incident. She was described as pretty much a saintly figure.
In Edmondo D'Amici's book Heart, one of the tales about "Heroic Italian Boys" has an adult version: Marco's mother Anna in 1000 Leagues to Find Mother. She has fallen gravely sick while working abroads and is extremely depressed as well, so she refuses treatment; when Marco finally finds her, Anna changes her mind and accepts to be operated on, which saves her life. The World Masterpiece Theater series based on this story keeps this plot point as well.
The title character of Banana Yoshimoto's novel Goodbye, Tsugumi. Young Tsugumi Yamamoto been ill all of her life, but still tries to live the rest at her fullest and keep interacting with those around her - specially the protagonist, her cousin Maria Shirakawa.
Katy's cousin and Cool Big Sis Helen is confined to her bed, and she's described as "a cripple". She's still a sweet and kind person whom Katy adores.
In a manner similar to Jill from Jack and Jill, Katy has an accident (this time she falls from a swing) and severely damages her back, thus spending the rest of the book in treatment. She's still not fully recovered in the second book.
Amy in What Katy did Next becomes very sick while travelling around.
Julia Valerian of The Mark of the Lion spends the second book dying of an ambiguous sexually transmitted disease, too penniless to even get it properly diagnosed, let alone treated before it is too far gone. Given her promiscuous, careless, and selfish behavior in the first book, when her plight is discovered, some characters see her as someone to be pitied; others think she’s getting her just desserts.
Colin Craven of The Secret Garden, who isn't actually ill, but is weak and gets sick a lot from spending all his time shut up in his room, never leaving his bed and having hysterical tantrums.
"The Boy" in The Borrowers, to whom Sho of The Borrower Arrietty is based on, had rheumatic fever — which, up to this day, was still pretty dangerous, as it causes inflammation... of the cardiac muscles.
Tony Makarios from The Golden Compass. His "illness" was that he had been forcibly separated from his daemon.
Callie's little brother Sam in Cut is a little cutie who has a severe case of asthma.
The main character of Donna Jo Napoli's "Breath"—subverted in that he doesn't really act the part, as he is very active throughout the story. (The epilogue to the story later explains that he has cystic fibrosis, and the story occurs during a time when nobody knew what that was, and it was a one-way ticket to an early and painful death.)
Also David's younger brother Bernard, who also led an astray life and is dying of tuberculosis in prison, lamenting how he couldn't fulfill a promise that he made to a child. David, as the Death Cart Rider, promises to fill that vow and helps Bernard to die in peace.
The Changeover: Jacko is being preyed upon by a sort of energy vampire, which prompts magical action on the part of his older sister.
Linton Heathcliff of Wuthering Heights is a cowardly, emotionally manipulative Spoiled Brat Ill Boy who constantly uses his illness to get what he wants, or at least to make other people miserable.
As mentioned above, while Janet Pecq from Jack and Jill suffered almost crippling damage to her back, her boyfriend Jack Minot got severe wounds to his right leg. his injuries aren't as bad as Jill's, but he also must go through lots of therapy and Character Development.
Jack's Big Brother Mentor Ed is slightly delicate, but nothing too big. But all of a sudden, towards the end of the book Ed gets so sick that hedies, and poor Jack suffers a HUGE Heroic BSOD when this happens.
The plot of Papelucho en la Clínica is kickstarted when Papelucho befriends Casimiro alias "Casi", a Lonely Rich Kid who's about to be operated on. They switch places for a prank, then Papelucho is operated instead by mistake and a scared Casimiro goes home to tell his dad about it. And halfway through the book, Casimiro has a relapse in his illness and is operated for real.
What Katy Did gives us Phil, Katy's only brother, in the fourth book (Clover). He and his sister Clover go to Colorado so he can properly recover.
Paul Strobe aka the lead male from Barbara Conklin's P.S I love you. And when he dies, he becomes the female lead Mariah's Lost Lenore.
Toklo's brother Tobi from Seeker Bears is first shown to be seriously ill and is too weak to talk, play, and even walk. In the sixth chapter of The Quest Begins, he finally dies and is shown to be very healthy as a spirit. Tolo's Story, however, shows that he wasn't always ill. Being swept away in a river (before being rescued by Oka) was the result of the ill cub we see at the beginning of the series.
Augustus and Isaac in The Fault in Our Stars, teenage cancer patients like Hazel. At the start Gus has lost a leg and Isaac an eye to the disease.
Jem Carstairs from The Infernal Devices, Has to take a drug to battle a disease he got from being totured as a child and coughs up blood if he goes to long without it.
Raistlin Majere from Dragonlance. He is a sickly child to begin with, and after his health is shattered in the Tower, he suffers from a disease that resembles tuberculosis or, since it is not contagious, a very dramatic version of asthma.
Renarin Kholin from The Stormlight Archive has a blood weakness that prevents him from fighting, which a source of angst for him, since he is part of the royal family of a warlike culture. In the second book, he says that he has epilepsy.