The statement said: "She was a much-loved mother, grandmother and great-grandmother and is survived by one younger brother.
"Winnie fought tirelessly for decades to find Keith and give him a Christian burial.
"Although this was not possible during her lifetime we, her family, intend to continue this fight now for her and for Keith.
"We hope that the authorities and the public will support us in this.
"We would like to express our deep gratitude to all the medical staff who cared for Winnie during her illness, and also the public for their good wishes and support over the years."
Twelve-year-old Keith was abducted on his way to visit his grandmother in Manchester on 16 June 1964.
He was one of five children murdered by Brady and his girlfriend Myra Hindley between 1963 and 1965. The couple were sentenced to life in prison in 1966.
Brady, 74, who has been detained at a psychiatric hospital since 1985, only confessed to killing Keith and his first victim Pauline Reade, 16, during an interview with a newspaper reporter 20 years later.
Mrs Johnson had been diagnosed with bowel cancer but had not given up her campaign, of almost half a century, to find out where her son's body is.
She had written to her son's killer several times. The last was a few months ago in what she said would be her final letter to him, telling him about her cancer.
Addressing him "Dear Mr Brady" she pleaded: "If you have got it in your heart to tell me where Keith is, it would end the trouble I have had for 47 years.
"I would like him found and buried before anything happens to me."
Before her death, she had been told about the alleged existence of a letter, apparently given by Brady to Jackie Powell, his mental health advocate, within the last month. The letter is understood to have been addressed to Mrs Johnson but Brady had said it could not be opened until after his death.
Police now believe the claim may have been a ruse orchestrated by the killer himself.