Is she the Catherine Wightman, age 53, in the 1860 census living at West Greenwich with Layton Hopkins? Probably.
She is probably the daughter mentioned as Catherine Whitman in the will of Jonathan Hopkins of West Greenwich dated 18 Sep 1848. This will also mentions land sold to Layton Hopkins. Thus, I have made an assumption that Jonathan is her father.
In the 1850 Federal Census, she appears to be Catharine Whitman age 45, living in Warwick, RI, with some of her children and her married daughter and son-in-law. Her son Henry and husband Thomas are in West Greenwich.
In the Federal Census of 1880, she is probably the Catherine Aylesworth, age 74, living with her son George W. Whitman and family in West Greenwich, RI.