In a book called Folktown with content from Éamon Phoenix and Sophie Rasmussen, they indicate this street was laid out in 1757 and led to Smithfield by 1791. It was named after Richard Barry, who according to Marcus Patton was a trustee of the [4th] Earl of Donegall (Patton 1993, pp. 31-32). The context was that the Earl was an absentee landlord living in England, considered mentally unfit to manage his affairs. As he had no children, his presumptive was his nephew, Arthur (his brother's son). An attempt was made in 1751 to have the Earl declared "a lunatic" by law and thus to have his title passed to his nephew, but this was rejected in court. These were the circumstances that led to Barry being appointed as a trustee to manage his affairs in Belfast (Benn i, pp. 533-34). He too was a nephew of the Earl, being the son of his sister, Lady Anne Chichester, and Lt.-Gen. James Barry, 4th Earl of Barrymore. He was born c. 1720 and died in 1787. He served as MP (Tory) for Wigan, 1747-61, succeeding his father.
It is referred to as Barry Street at that time in correspondence (Benn 1877, p. 540). Later it was marked as Berry Street on Mason's 1815 map, and this slightly corrupted form persisted. In between, it was also referred to as Factory Row. This may reflect a difference between an official name and a more popular name, which may have both existed in parallel. Berry Street is included in the list of streets in the Belfast Directory 1831-1832.
Again according to Patton, the General Dispensary was set up here in 1792; it was then transferred to a larger premises to become the Belfast Fever Hospital and General Dispensary in 1797. It then moved again to 1799 to West Street.