Elizabeth was the 4th daughter of Thomas Strangways, of Melbury Sampford, Dorset, and his wife Susan Ridout, and was baptised 20 February 1691 at Church of St Mary, Melbury Sampford, Dorset, though it's not clear whether this is 1690/91 or 1691/92.
She married James Hamilton, 5th Duke of Hamilton, as his second wife, in 1727, but died without issue on 3 November 1729 on the road from Bath. She was buried on 13 November 1729 at Melbury Sampford, probably in the Church of St Mary.
Cokayne, George Edward (ed.), The Complete Peerage: or a history of the House of Lords and all its members from the earliest times, 2nd ed., vol. 6, revised and edited by Hon. Vicary Gibbs, H.A. Doubleday, Duncan Warrand & Lord Howard de Walden, London: St Catherine Press 1926., p. 270. Digitised by ABC Publications, 2003.
"England, Dorset, Parish Registers, 1538-1936," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:QJDH-NF2Q : 24 July 2016), Christening: Elizabeth Strangways, 20 Feb 1691, dau Thomas & Susanna Strangways, citing Melbury-Sampford, Melbury-Osmond, Melbury-Sampford, Melbury-Osmond, Dorset, England, Record Office, Dorchester; FHL microfilm 2,427,538.
"England, Dorset, Parish Registers, 1538-1936," index, FamilySearch (https://familysearch.org/pal:/MM9.1.1/QJDH-NPMY : accessed 30 Oct 2014), Elizabeth Hamilton And Brandon, buried 13 Nov 1729; citing Melbury-Sampford, Melbury-Osmond, Dorset, England, Record Office, Dorchester; FHL microfilm 002427538.