St. Mary's County Marriages does not show a marriage of James French to either woman.
For Elizabeth Meekin:
Elise Jourdan, Colonial Recors of Southern Maryland
Mary Louise Donnelly, "The French Family:, Colonial Settlers of St. Clement's Bay, 1634-1780
Colonial Families of Maryland: Bound and Determined to Succeed By Robert William Barnes, at page 93
There is a very well-argued, but inconclusive essay by Mara French at: http://www.frenchfamilyassoc.com/FFA/CHARTS/Chart137/#_James_French...
She writes: "According to the inventory of Luke Gardiner taken in 1674, James French was one of his servants on S. Clement's Island (Inv 1:111). When he completed his term as a servant, James French married Elizabeth Meakin, the daughter of William Meakin and Margaret Beard. It was through this marriage that he became the possessor of 129 acres of "Wolver Hampton" which bordered on "Hopton Park". After William Meakin’s death, his tract totaled 150 acres and he later received 100 acres of "Hopton Park" from John Baptist Carberry, an Irish immigrant. James French and his descendants lived at "Wolver Hampton" until after the Revolution . According to Wikipedia, Wolverhampton is a city and metropolitan borough of the West Midlands in England. We don’t think James French came from that area, but so far we have no proof.
"We need more substantial evidence to prove this marriage. This was probably based on the fact that William Meakin patented "Woverhampton", 150 ac. in Newtown 100, 5/3/1681 which was possessed by James French in 1707. Land was sometimes sold, not always inherited! I find no proof that the wife of James French was Elizabeth Meakin! See the bibliography for the Ancestry of the William Meakin Family.
"Elizabeth Meakin d. after 1733, as the widow of James French. This record has not been found."
http://familytreemaker.genealogy.com/users/m/c/c/Vickie-S-Mccracken... (spouse of James French, but does not mention Mary French)
This site raises a third alternative that Mary Brown is Mary French Thompson's mother: