"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).
Perhaps it is most likely to 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. See also Chichester Street.