In Chapter 11 of To Kill a Mockingbird, Jem and Scout Finch encounter Mrs. Dubose, their cranky and unpleasant neighbor. When Mrs. Dubose insults Atticus, Jem exacts revenge by knocking the tops off of every one of camellia bushes; as punishment, he is required to read to Mrs. Dubose each school day and Saturday for two hours. Scout, who feels it is her duty to look out for her brother, accompanies Jem as he goes to read to Mrs. Dubose.
During their reading sessions with Mrs. Dubose, Jem and Scout are puzzled by her behavior. After a certain amount of time, during which Mrs. Dubose corrects Jem's reading and torments him, Mrs. Dubose appears to slip into an almost comatose state. Soon afterward, her alarm clock would sound and the children would be dismissed by Jessie, Mrs. Dubose's caretaker.
One evening, Atticus receives a phone call and leaves for Mrs. Dubose's home. When he returns, he presents Jem with a box containing a camellia bloom and tells the children that Mrs. Dubose has passed away. Then, he explains the cause of her strange behavior.
"Mrs. Dubose was a morphine addict," said Atticus. "She took it as a pain-killer for years. The doctor put her on it. She'd have spent the rest of her life on it and died without much agony, but she was too contrary--...Just before your escapade she called me to make her will. Dr. Reynolds told her she had only a few months left. Her business affairs were in perfect order but she said, 'There's still one thing out of order.'...She said she was going to leave this world beholden to nothing and nobody. Jem, when you're sick as she was, it's all right to take anything to make it easier, but it wasn't all right for her. She said she meant to break herself of it before she died, and that's what she did...Most of the time you were reading to her I doubt if she heard a word you said. Her whole mind and body were concentrated on that alarm clock..."
When Mrs. Dubose was able to endure an entire two hour reading session without needing a dose of morphine, she knew that she had finally beaten her addiction. Jem continued to read to Mrs. Dubose for another week, during which time her improvement was evident; at the end of that week, she released him from his obligation. Through sheer will power and with a little bit of distraction, Mrs. Dubose was able to die peacefully and free.
Mrs. Dubose uses the alarm clock to mark the time between her doses of morphine. She is a drug addict trying to wean herself off of the painkiller before she dies. The alarm clock tells her when she can have the next dose, and the time before the alarm gets a little longer every time to increase the time she is able to go without morphine. The children read to her to distract her from the pain without the medication.
If you read the summary of Mrs. Dubose's character, it says that reading helped her overcome her morphine addiction. The alarm clock was simply a part of this process.
Mrs. Dubose is a morphine addict. The drug was probably prescribed as a painkiller for some illness she contracted. One side effect of morphine is irrationality and paranoia. Her lashing out at Jem for Atticus's defense of Tom Robinson was not a rational act. After Jem destroys her camellias in retaliation for her racial slurs on Atticus, his punishment is to read to her daily. What Jem does not know is that the purpose is two-fold. Jem must see her as a pathetic, old woman no longer in control of her own body and mind. The real purpose is to wean her from the morphine, so that she can die with a clear mind. The alarm clock has minutes added to it daily, not to punish Jem, but to increase her time for not depending on the drug. The longer she can go without it, the more her mind will clear.