I found the following information about Sir Richard Cox and his family. This seems to refute the claim of those that say that John Cox (1694-1747), husband of Mary Rankin, was the son of Sir Richard Cox, as John, son of Sir Richard died in 1692.
A Genealogical and Heraldic History of the Landed Gentry of Ireland
Family of Cox
Sir Richard Cox, 1st bart. of Dunmanway, Lord Chief Justice of Common Pleas in Ireland, was appointed 1703, Lord High Chancellor of Ireland, and was twice one of the Lords Justices during the absence of the Lord-Lieutenant. He was b. 1650, knighted 1692, and created a baronet 21 Nov. 1706. He m. 26 Feb. 1673, Mary Bourne, by whom (who d. 1 June, 1715) he had, with seven daus, five sons.
1. Richard, M. P. for Tallaght 1705, and for Clonakilty 1717, b. 27 Oct. 1677; m. 1st, 5 Sept. 1698, Susannah French, who d. 10 Aug. 1716, and 2ndly, 13March 1719, Mary, dau. of Very Rev. Arthur Pomeroy, Dean of Cork, and d.v.p. 15 April, 1725, leaving issue by his 1st wife only (with four daus.), three sons,
1. Richard (Sir), 2nd bart. of Dunmanway, M. P. for Clonakilty, b. 23 Nov. 1702; m. 13 Dec. 1725, Mary, dau. of Right Hon. George Evans, and sister of George, 1st Lord Carbery, which lady d. Lan. 1768. He d. Feb. 1766, having had (with four daus, the eldest of whom, Mary, m. Joshua Hamilton, and had a son, Henry Hamilton, who assumed by Acto of Parliament the surname of Cox), three sons,
1. Richard, m. Eliza Turner, dau. of John Becher, of Creagh co. Cork, and d.v.p.
2. Michael (Rev. Sir), 3rd bart. of Dunmanway, m. 7 Jan. 1762, Hon. Elizabeth Massey, dau. of Hugh, 1st Lord Massy, and widow of John Arthur, of Seafield, co. Dublin, and d. 18 July 1772, leaving (with one dau.) two sons,
1. Richard Eyre (Sir), 4th bart. of dunmanway, m. 7 Jan. dau. of Edward O'Brien, grandson of William, 3rd Earl of Inchiquin, and sister of William and James, and 3rd Marquesses fo Thomond, and was accidentally drowned 16 Sept. 1784, leaving by her (who m. 2ndly, the Right Hon. William Saurin, Attorney-General for Ireland) an only dau, Maria.
2. Michael, d. unm. before his brother.
3. George, d. unm.
2. James, b. 13 Dec. 1703; d.s.p.
3. John, of Bandon, b. 14 April, 1705, had (with four daus.) three sons, of whom, Richard, the eldest, Capt. 46th Regt., d. at the Havannah; John, the 3rd, Major H. E. I. C., d.s.p., and the 2nd, Michael, Lieut.-Col. in the Guards, m. Jan. 1765, Anna Maria, dau. of Daniel Shea, a West Indian planter, and d. 4 Feb. 1782, leaving by her (who d. Nove. 1793), with four daus, four sons,
1. Richard (Sir), 5th bart. of Dunmanway, b. 6 June, 1769, lost in his passage from Bengal, Sept. 1786, d.s.p.
2. John (Sir), 6th bart. of Dunmanway, an Officer in the Life Guards, b. 4 April, 1771; d.s.p 23 Dec. 1832.
3. Michael Joseph, Capt. 1st Royal Scotch Regt.; d.s.p. at Antigua 1804.
4. George Mather (Sir), 7th bart. of Dunmanway, Major-Gen. Bombay Army, b. 24 Feb. 1777; d.s.p 28 June, 1838.
2. Walter, b. 8 Feb. 1679; d. 5 June, 1683.
3. John, b. 22 April, 1684; d. 2 Dec. 1692.
4. William, b. 1 Jan. 1686; d. Aug. 1693.
5. Michael (Most Rev.), D.D., Archbishop of Cashel, b. 2 Nov. 1691; ordained Priest 20 Sept. 1713; Chaplain to James, and Duke of Ormonde, when Lord-Lieutenant; consecrated Bishop of Ossory 29 May, 1743; translated to Cashel 22 Jan. 1754; m. Anne, dau. of Hon. James O'Brien, M.P. for Youghai, and grand-dau. of William, 3rd Earl of Inchiquin, by whom (who d. 19 Jan. 1745) he left at his decease, 28 May 1779, an only son,
1. Richard, of Castletown, co. Kilkenny, b 15 Jan. 1745; m 25 Jan 1766, Mary, dau. of Francis Burton, brother of Sir Charles Burton, 1st bart. of Polacton, and had (with two daus., 1, Rachel, m. Ponsonby Hore; and 2, Anne, m. Hon. Price Blackwood) five sons,
1. Michael, of Castletown, b. 14 April, 1798; m. Hon. Mary Prittie, dau. of Henry Sadleir, 1st Lord Dunalley, and had issue, (Note: Michael's date of birth seems off, was there a typo and the date was actually 1768?)
1. Richard (Sir), 8th bart. of Dunmanway, d.s.p. 7 May, 1846.
2. Henry, d. unm.
1. Katherine, heir to her brother, m. 1 June, 1833, Lieut.-Col. William Villiers Stuart, M. P. co. Waterford, afterwards of Castletown.
2. Francis (Sir), 9th bart. of Dunmanway, b. 23 July 1769; m. Aug. 1803, Anne Maria, dau. of Sir John Ferns, and d.s.p.m. 6 March 1856.
3. Richard (Rev.), Rector of Cahirconlish, b. 20 Sept. 1771; m. Sarah, dau. of Rev. Ralph Hawtrey, and had
1. Hawtrey (Sir), 10th bart. of Dunmanway, b. 1808; m. 1857, Elizabeth, widow of Henry Hore, and d.s.p. 12 April, 1872.
2. William Saurin, m. Mary, da. of Major Miller, of co. Derry, and had, Richard, d. unm. v.p; Margaret.
3. Michael (Sir), 11th bart., d. unm. 15 June. 1872.
4. Francis Hawtrey (Sir), 12 bart. of Dunmanway, m. 6 March, 1853, Emma Katherine, dau. of duncan McKellar, and d.s.p. 17 Oct. 1873, when the baronetcy became extinct.
1. Maria. m. Rev. Matthew Enraght.
2. Anne, m. Rev. Thomas Lyon.
4. William, b. 6 Jan. 1773; d.s.p. in the West Indies before 1811.
5. Benjamin, b. 20 Jan. 1775; d.s.p.
Sir Richard Cox, the 1st bart., Lord Chancellor, d. 3 May, 1733, aged 83 years.