Yes, it is the descendants of Thomas Carrington.
Looking through, I see where it says that all descendants of Samuel Carrington (father of Thomas) are eligible to be members of the Daughters of the American Revolution. The D.A.R. lineage for the descendants of Alexander Carrington is:
Samuel Carrington 1754-1818
Thomas Carrington 1782-1879
Asa Carrington 1818-1888
Alexander Carrington 1839-1919

Reading from this book.....The name of Carrington is believed to have derived from the residence of its first bearers, in Carrington, Chester County, Egland. It is found in ancient English and early American records in the variant spellings of Carrynton, Carrinton, Carinton, Carynton, Caringron, Carrington, and others, of which the form last mentioned is that which is most generally used in America today. Families of this name were residents at early dates in the English counties of Chester, Lincoln, York, Hertford, Essex, Buckingham, Leicester, Wawick, and London.