Henry M. Arderne was the elder son of Ralph Henry Arderne (1802-1885), a timber merchant and member of the Cape Legislative Assembly. He qualified as a solicitor at University College, London, was admitted as an attorney in the Cape Supreme Court in 1857, and became a partner in the firm Fairbridge & Arderne. For most of his life he lived on the estate "The Hill" in Claremont (a municipality since 1886, but from 1913 part of Cape Town) which his father had developed from 1845. He extended and improved the estate, which became widely known for its fine collection of exotic and indigenous trees and plants. It became the Claremont Public Garden in 1927 and was renamed the Arderne Gardens in 1961.
Arderne was an active plant collector and in 1886 discovered a white watsonia on Roman River Farm near Tulbagh which he cultivated in his garden. It was named Watsonia ardernei in his honour. He sent plants to the botanic gardens at Kew, and in July 1885 was a founding member and served on the first committee of the City and Suburban Horticultural Society. He was also a member of the South African Philosophical Society from its foundation in 1877 to 1906, and a trustee of the South African Association for the Advancement of Science from 1903 to at least 1913. In 1897 he led a group of persons, the so-called Arderne party, to Bulawayo for the opening of the Northern Extension Railway, and in 1904 the party travelled on the first through passenger train to the Victorial Falls. He visited Europe frequently, was a fellow of the Royal Horticultural Society, and delivered an address on the history and contents of his garden which was published in the society's Journal in 1902.