He was the oldest son of the judge Sir Gilbert Elliot, 1st Baronet, of Minto (c. 1650–1718).
[1] Elliot studied law at the University of Utrecht and was admitted to the Faculty of Advocates in 1715.
[2] In June 1726, he was made a judge of the Court of Session, taking the judicial title Lord Minto.
[3] In 1718, Elliot was married to Helen Stewart, the daughter of Sir Robert Steuart, 1st Baronet, who had been a member of the pre-union Parliament of Scotland.
Through his eldest daughter Eleanor, he was a grandfather to John Rutherfurd, MP for Roxburghshire (who married Mary Ann Leslie, the only child of Maj.-Gen. Alexander Leslie), Elizabeth Rutherfurd (who married Andrew St Clair of Herdmanston, de jure 12th Lord Sinclair, parents of Charles St Clair, 13th Lord Sinclair)[8] and Jane Rutherfurd (who married William Oliver of Dinlabyre).