Actually, 'Madras' was just a short form of a fishing village that existed near Royapuram.
'Chennai', is actually a Telugu name. The British got the land for setting up their fort and the surrounding neighborhood (The area around Madras Fort station) from a local general (based out of Sri Kalahasti on the Gudur-RU line in AP). At the time of writing the deed, the general requested that the new township be named after his father, Damerla Chennappa Nayakudu (Hence the name Chennai), but the name 'Madras' just stuck around as people in the region were probably more used to calling it by that name.