'Abaya b. Rib'i asked 'Abd Allah b. al-'Abbas: "Why did the Prophet (s) give the title "Abu Turab" to 'Ali (a)?" He replied: "Because he [Ali (a)] is the owner of the earth and God's representative among people. The earth continues to exist and remain calm because of his presence. I heard the Prophet (s) saying: 'In the Hereafter, the non-believer sees the position and the reward of the followers of Ali's and says: 'I wish I were of the dusty' (so I was related to the father of dust ['Ali (a)] and was one of his Shi'a) and this is what Qur'an reads 'and the faithless will say 'I wish I were dust' (Quran 78:40).'"
Katwari says: "The first person titled as 'Abu Turab' was 'Ali b. Abi Talib (a). When the Prophet (s) saw him laying on the ground and saw the dust on his clothing and body, he gently and kindly said to him: 'Get up, O' Abu Turab!' and 'Ali b. Abi Talib (a) liked this title the most. After this incident and because of the Prophet (s), it became a special feature for him, that the dust (earth) told him the news of the past and the future to the Day of Resurrection."
In a different hadith, it is said: "In the middle of the month of Jumada I 2/November 623, the Prophet (s) and a group of his companions left Medina seeking the caravan of Quraysh led by Abu Sufyan. They reached a place named 'Ashira but they did not find the caravan. During their stay in 'Ashira, the Prophet (s) encountered 'Ali (a) and 'Ammar while they were asleep on soil and there was dust over Ali's face and head. The Prophet (s) gently and kindly woke them up and said to 'Ali (a): O' Abu Turab! Get up. Do you not want me to tell you who are the most vicious people on earth? One of them is Ihmir who killed the she-camel of the Prophet Salih (a) and the other is the one who hits you on the head by a sword and dyes your beard with your blood.'" From the time 'Ali (a) became famous as 'Abu Turab'... Ali loved this title; because the Prophet (a) had given it to him.
According to the last tradition, the literal meaning of "Abu Turab" (i.e. the father of dust) may not be an exact meaning of "Abu Turab" as a title but rather the term "dusty person" would convey the meaning better.
During the Umayyad Period
Umayyad Caliphs issued that every speaker in any Muslim city should curse 'Ali (a). In this period, the title "Abu Turab" was used as a derogatory title for Imam Ali (a).