Atticus believes this because of how bravely Mrs. Dubose fought against the disease that was killing her.
Mrs. Dubose was dying and had been using morphine to dull the pain from her disease. But, towards the end of her life, she chose to stop using the morphine. She wanted to just take the pain and live with her mind clear rather than continuing to use the drugs.
Atticus says that it takes a brave person to keep fighting even though they know they have lost, and that is exactly what Mrs. Dubose had done.
In some ways, Atticus sees some of himself in Mrs. Dubose. Though he has not dealt with the ravages of a morphine addiction, he likely knows that his defense of Tom Robinson is similarly a painful and eventually a futile effort.
And this is what he most respected in Mrs. Dubose. She knew that the many years of addiction to morphine had killed her and knew that she wouldn't survive the withdrawal from the drug. Yet she chose to fight through it anyway in order to have some clarity. The difficulty of this was tremendous, particularly in a time when there were no other drugs to ease the withdrawal.
Atticus' effort to defend Tom is also deeply personal, dangerous at times, and Atticus has a clear understanding of the difficulty of convincing any jury that Tom is not guilty. Yet he takes on the task anyway because it is the right thing to do. The type of bravery he wishes to have, Mrs. Dubose already had and demonstrated. One more reason why he tells the children that he considers her one of the "bravest" people he'd ever known.
Mrs. Dubose was one of the meanest women in Maycomb County, according to Scout. She and Jem had plenty of run-ins with her, and the two of them didn't like her very much. What the children didn't know was that Mrs. Dubose was a very ill woman. She had suffered for a long time. He doctor had prescribed morphine for her to help with the pain, and she eventually became hooked on it. She was slowly dying and Atticus knew what her plan was before she died. When Jem had to go and read to her, he was an important player in her plans.
Atticus thought of Mrs. Dubose as the bravest person because although she was in pain and dying, Mrs. Dubose was determined to break her habit. She faced death with strength and determination. She didn't blame anyone, she just knew what she had to do, and she did it.
Atticus had so much respect for the woman. Yes, she was mean and nasty, but she had been through so much. She had been on morphine for many years and that had affected her moods. She was a strong woman who was willing to face her own demons before dying, and that is exactly what she did. To Atticus, that made her the bravest person he knew.