We don't really know what type of cancer Lillian has in Okay For Now. In fact, the word 'cancer' is never mentioned in the story.
According to the text, we initially discover that Lillian is sick when she cannot play the part of Helen Burns in the Broadway play. At the end of Chapter Nine, Doug finds out that Lil has been admitted to the hospital. He thinks the hospital visit is due to the stomachaches Lil has been getting on account of her pencil-chewing habit. However, Lucas mysteriously informs Doug that Lil hasn't been admitted for a simple stomachache.
At the beginning of Chapter Ten, Mr. and Mrs. Spicer have been given a statistic: 'One in four.' This statistic represents a one in four chance of survival for anyone having the disease Lil has. When Doug visits Lil at the hospital, she is hooked up to what appears to be an IV (intravenous) drip. An IV drip lets fluids and medicines be fed directly into the bloodstream. During Doug and Lil's conversation, a nurse comes in to draw some blood from Lil. Although at this point, the word 'cancer' is still not mentioned, many clues in the text lead us to infer that Lil is suffering from some sort of cancer.
Many cancer patients need to have blood drawn before undergoing chemotherapy. This procedure allows doctors to assess blood count levels before treatment. If blood counts are low, doctors can adjust chemotherapy dosages or delay the treatment until blood count levels have stabilized. Some cancer patients also need to have blood drawn periodically so that doctors can assess the effectiveness of any experimental treatment on the patient. In the story, Lillian has lost her hair, a side-effect from chemotherapy. Read about a cancer patient's experience with frequent blood draws.
Although we cannot know for sure why the author chooses not to give a name to Lil's disease, we experience a surge of hope in reading the last lines of the novel:
...I heard all around us, over the sounds of the huge machines in the room, over the sounds of Apollo 11 heading to the moon, I heard, all around us, the beating of strong wings.
The story of Lil's illness and her relationship with Doug illustrates the purpose of enduring courage, the beauty of fidelity in the face of suffering, and the motivation to overcome against all odds. I believe the author wanted us to concentrate on the beauty and resilience of the human spirit instead of the destructiveness of disease. Hope this helps!