Shanidev and Lord Hanuman
Lord Hanuman is said to have rescued Shani from the clutches of Ravana in Ramayana and in gratitude Shanidev promised to reduce his ill effects on Hanuman devotees especially if they pray to Lord Hanuman on Saturdays.
There is another version where it is said that when it was time to influence the life of Lord Hanuman Shani climbed on to Hanuman's shoulder and that a fearless Hanuman immediately assumed a large size, and Shanidev was caught painfully between Hanuman's shoulders and the ceiling of the room they were in. As the pain was unbearable, Shani requested Lord Hanuman to release him and in return promised to reduce the malefic effects of Shani for those who pray to Lord Hanuman.
Thus worship of Lord Hanuman is suggested for reducing the harsh effects of the unfavourable presence of Lord Shani.