My understanding is that Barnwell was derived from "DeBarneval" a name of French origin. The family was allied with to the Dukes of Little-Bretagne, where the name still continues. it was established in Ireland in 1172 by Sir Michael De Berneval, one of the companions in arms of "Strongbow" the Earl of Pembroke. Over time, variations in the spelling of the name occured in Ireland with D'Barnable, Barnhwall, Barnhell, and Barnehell.Later changes included Barnwell, Barnhale, and Barnhill.
My wife's line is from "Big Robert" Barnhill who came to Chester County, Pennsylvania from Northern Ireland about 1765. In 1776, Big Robert and family moved to Orange County, North Carolina. About 1780, the family name was changed to Barnwell. The family story is that the name change was to aviod confusion with several other Robert Barnhills living nearby. However, my hunch is that the name change occured because 2 of Robert's sons, James and John Barnhill, served under Col. John Barnwell (from South Carolina) during the Revolutionary War.
My wife's line line then moved to Cobb County Georgia, and then to Morris County Texas.