The street was named in honour of Henry Peter, Lord Brougham (1778-1868) famous lawyer and Whig (Liberal) politician (John J. Marshall, Belfast Telegraph, 03/03/1941). Britannica describes him as "a noted orator, wit, man of fashion, and an eccentric". He was a fierce opponent of slavery and played a leading role in establishing the University of London as the first non-denominational third-level institution. He was Lord Chancellor of England 1830-34. He also designed the Brougham carriage, a versatile four-wheeled carriage pulled by a single horse, built in 1838 by coach-builders Robinson & Cook. In this sense the word brougham entered the dictionary as a common noun, first recorded in the 1850s. It may be coincidence, but one end of Brougham Street is close to the site of the former York Road Railway Station, where brougham carriages may have waited to bring train passengers to the city in the late 19th century. Note that the date given for Lord Brougham's death by Marshall is incorrect.
Brougham Street is first recorded in Martin's Belfast Directory, 1841. Brougham Street (Left side of York Street) is featured in the 1850 edition of Henderson’s Belfast Street Directory.