Boswell met a Mr. Mayne from Scotland, probably on November, 1762, in London. Apparently Mr. Mayne had introduced Boswell as a member of the Society for the Encouragement of Arts and Sciences in 1760. Boswell, however, had not paid his dues ever since, which Mayne told him.
The London Journal tentatively identifies Mr. Mayne as a John Mayne, merchant. It has not been possible for me to get any futher information with any certainty.
Curiously, however, the widow of a John Mayne (d. 1785) of Teffont Ewyas (in Wiltshire), Isabella Raymond (d. 1808), married Archibald, 9th Earl of Dundonald, a distant relative of Boswell's, in 1788. Archibald was the son of Thomas Cochrane, Boswell's great granduncle.1 This may be the Mayne from Boswell's journal - or it may just be a co-incidence.
Note 1: Information about John Mayne from History of the Commoners of Great Britain and Ireland. (Mayne, of Teffont Ewyas)