Conyngham, Henry (1705/6–81), 1st Baron Mount Charles , Viscount and Earl Conyngham, and MP, was the posthumous son of Maj.-gen. Henry Conyngham, who died in battle in Spain 1705/6, and Mary Conyngham (née Williams), who was previously widow of Charles Petty, 1st Lord Shelburne. The family had estates at Slane, Co. Meath, and Mount Charles, Co. Donegal. Henry Conyngham served as an army officer for twenty years until his elder brother's death in 1738.
He was MP for Killybegs, Co. Donegal, in the Irish parliament (1727–53), but devoted most attention to his parliamentary career in England and to his estates there. He represented Tiverton (1747–54) and Sandwich (1756–74), and generally supported the administrations in power; there is no record of any speech in parliament. He was created Baron Conyngham of Mount Charles on 3 October 1753, but never took his seat in the lords; he sought further ennoblement, and was created Viscount Conyngham (20 July 1756) and Earl Conyngham (4 January 1781). His Irish peerages were no impediment to his sitting in the English house of commons. Conyngham was vice-admiral of Ulster 1748–79, and unsuccessfully sought other posts on the Irish establishment.
Recent research has shown that earlier authorities erred in stating that Henry Conyngham excavated the Roman circus at Tarragona in Spain; it seems certain that it was his nephew, William Burton Conyngham (qv), who carried out this pioneering archaeological excavation. Henry Conyngham married (December 1744) Ellen Merrett, daughter of a London merchant. When Conyngham died childless on 4 April 1781 at Bath, his estates in Donegal and Meath went to William Burton, who took the surname Conyngham; the titles and other estates were inherited by Francis Pierpoint Burton, William Burton's elder brother, who likewise adopted the surname Conyngham in 1781. They were the sons of Francis Burton of Buncraggy, Co. Clare, and of his wife Mary (née Conyngham). Sir Francis Pierpoint Burton Conyngham (c.1721–87) was MP for Killybegs (1753–60), and for Co. Clare (1761–76). He married (19 March 1750) Elizabeth, daughter of Nathaniel Clements (qv); they had two sons and three daughters. He was known for his parliamentary independence, although Clements may have had some influence over him. He spent much of his time in England and died 22 May 1787 at the Hot Wells, Bristol. He was succeeded by his son, Henry Conyngham (1766–1832), who became 1st Marquess Conyngham (1816).