"Arthur Street, so called from the rather curious fact that there were five Earls of Donegall in reregular succession for more than 150 years (1647 to 1799) all named Arthur Chichester, in honour of one of whom this street received its name" (George Benn, A History of the Town of Belfast, vol. i, 1877, p. 529-30).
It could be named after the 5th Earl who held the title in 1790 when the street-name is first recorded. He was created Baron Fisherwick in this year and created Marquess of Donegall and Earl of Belfast in the following year.
However, Marcus Patton suggests that is probably named after Hon. Arthur Chichester (1771-88), son of the 1st Marquis of Donegall (Patton 1993, p. 18). If so, it would be the first of "sibling set" commemorating Arthur and his sisters (Henrietta Street, Charlotte Street, Eliza Street, Amelia Street) all of whom died either in infancy or relatively young. The dating favours this interpretation as the street was named 2 years after Arthur's death.
See also Chichester Street.