Imam Tirmidhi's Ism (proper name) was Muhammad. His Nasab (full name including his genealogical chain) was: Muhammad b. 'Isa b. Thawra b. Musa b. al-Dahhak. Imam Tirmidhi's Kunya (honorific name) was Abu 'Isa. Certain scholars raised objections regarding his kunya, since there is a tradition in the Musannaf of ibn Abu Shaiba, (a particular type of Hadith book) where Prophet Muhammad (SAW) reprimanded someone whose name was Abu 'Isa by saying that Isa had no father, implying that it was not becoming to keep the kunya Abu 'Isa. Hence the question arises regarding Tirmidhi's doing so. The permissibility for this practice is established in Sunan Abu Dawud, where it is narrated that Muqhira b. Sh'uba (d.50 A.H.) who was a companion of the Prophet (SAW), had the kunya Abu 'Isa. Once, 'Umar b. Khattab (d.24 A.H.) the second Caliph of Oslam rebuked him by saying, "Does it not suffice you to be called Abu 'Abd AllAh". Mughira replied, "It was the Prophet (SAW) of Allah who gave me this kunya". This incident proves that the prohibition was abrogated since the Hadith in which the Prophet (S.A.W.) prohibited the usage of the kunya was prior to the one in which he gave permission for its usage, hence the permission coming after the prohibition abrogates it.
Moulana Muhammad Yusuf Binnawri (d.1397 A.H.), a commentator, on the Jami', said that Imam Tirmidhi was born in Bugh in the year 209 A.H. He belonged to the Banu Sulaym tribe, hence he was called Sulami. His native town Bugh is a few miles from Tirmidhi and is considered a suburb of Tirmidh. Therefore he was known as Tirmidhi and Bughi, both places being in Khurasan, which is presently a province in North Eastern Iran, and it is with the nisba (linking him to his place of origin) Tirmidhi, that he is well known. The aforementioned biographer of Imam Tirmidhi does not mention any details of his parents. Imam Tirmidhi is reported to have said that his grandfather belonged to Marw but he subsequently moved to Tirmidh. Likewise, no mention is made of Tirmidhi's early life or occupation.