I looked into this a little more, because I wanted to see the surname "Cutler" in the ancestry of Thadeus Cutler Gilbert. I haven't had time to find the records yet - but I did quickly find online posts by descendants of Thadeus's parents who refer to them as Asa Gilbert and Hannah (Cutler)(Barnes).
So Hannah was likely born Hannah Cutler and then married a Barnes and then married Asa Gilbert. The record of her marriage to Asa would then give her name as Hannah Barnes, and that is why so many family trees list her that way.
Alternately, she could also have been born Barnes and then married Cutler, I suppose. The truth of it will have to be found in the original records.
A poster on the Gilbert Family Forum, Genforum, mentioned that the Cavendish Historical Society, Cavendish, Vermont has published a book of area genealogy called "Families of Cavendish." Surnames covered in the book include Gilbert.Another poster asked, "Would it be at all possible for you to check the Cavendish book for Asa Gilbert and Hannah Cutler who were married in Brookfield, Mass., 3 Jan 1771. They later removed to Cavendish."
Asa and Hannah had a son Asa b. 1879. I'll call the son Asa junior for convenience - he wasn't called that really.
Another thread in Gilbert Family Forum, Genforum mentions this Asa M. Gilbert (junior), the brother of Thaddeus Cutler Gilbert b 1781. Asa junior (baptized 03 October 1779 at Brookfield, Massachusetts) married in Saratoga, New York to Hannah Johnson. In about 1826 Asa junior and Hannah moved to Washtenaw County, Michigan. Asa junior died in 1828 leaving Hannah with 10 children. The poster did not name sources. Here's a link to the discussion, which includes several descendants, LINK: http://genforum.genealogy.com/gilbert/messages/68.htmlhttp://genforum.genealogy.com/gilbert/messages/68.html
What a fun family history! You apparently have many Gilbert cousins who have worked on the family history and would likely be pleased to hear from you.