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Ann White, who married Richard Brown Bamber, was the daughter of Archibald White of Dublin (d. abt. 1735) and his wife Jane Smith. Archibald's will names his wife Jane White otherwise Smith. Three daughters Margaret, Martha and Ann. Cousin James White, exor. Cousin Timothy White, exor., brother to said James White. Cousin Archibald White son to his uncle Archibald White. Henry Lecky, Coleraine, Co. Londonderry, Esq., and Wm. Hamilton, Dunnemanagh, Co. Tyrone, trustees. Robert Dent, Dublin, gent. Money due to him out of residuary assets of James Martin. Town and lands of Derryfore [Queen's Co.].

A transcript of Archibald's will can be found here:
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSH3-3C3Q?i=588&cat=185720

The lands of Derryfore came to Ann and were part of her marriage settlement which can be seen in the following deeds:

https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSJW-GD85?cat=185720
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSH3-M7FK-P?cat=185720

Archibald was the son of John White/Whyte, the younger, merchant of Belfast. He was the brother of Fulke White of Whitehall. John's family is very well documented in Jean Agnew's Belfast Mechant Families in the Seventeenth Century, p.250-52.

Jane Smith (Archibald's wife and Ann's mother) was the daughter of Andrew Smith (d.1700) and his wife Martha Bell. Martha's brother was the alderman, Thomas Bell (and Mayor of Dublin 1702-03). Jane's grandfather was Samuel Smith of Belfast, (an abstract of his 1726 will can be found at findmypast.com).
 
See the following:
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSHW-ZCXJ?cat=185720
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSH3-M3BN-7?i=473&cat=185720
https://www.familysearch.org/ark:/61903/3:1:3Q9M-CSH3-9HQ9-X?cat=185720