You've no doubt found the family in the 1860 census of Marathon, Lapeer County, Michigan, which has:
Abel Beckwith, 57, farmer, b. NH Carlindena Beckwith, 56, b. VT Lawson Beckwith, 14, b. NH and then in the same household the family of George Beckwith, 35, Laborer, b. VT Harriet Beckwith, 24, b. NH and four children.
Given the son named Lawson, it is interesting that the 1850 census of Swanzey, Cheshire County, NH has a Lawson Beckwith, 37, Innkeeper, with wife Eleanor and three children, the oldest 10, all born NH.
Your Abel is quite likely the Abel Beckwith who is in the 1840 census of Addison County, VT as:1 male 30 < 40; 1 female 30 < 40; 1 male 15 < 20; 1 male 10 < 15; 1 male 5 < 10; 2 females < 5.
Many of the people who lived in Addison County came from New London or Lyme or East Haddam, Connecticut.No doubt Abel's family originated from down there, but we know that he was born in New Hampshire.And I find (although one cannot always believe what one finds there) on the web anAbel Beckwith who married 23 March 1759 Lucy Calkins in Lyme, CT, and she died in Marlow, Cheshire County, NH!! He was conceivably your Abel's grandfather?