I am currently researching Almira C. Strang and maybe what I have found might shed some light on your research although I can't say I have concrete answers.

I, too, cannot find a birth or marriage certificate for Barney, but I have found Barney and Almira in the 1860 Massachusetts Census. George W. Thompson is listed in that record. (You've probably seen it). I came across an enlistment for Barney in the Civil War and wonder if he was killed in that war?? Almira's marriage to Andrew J. Thompson (I don't think he is related to Barney?) is her second marriage as stated in that marriage record.

Hopefully that sheds some light on things, at least why she is found with Andrew J. Thompson instead of Barney. Now if we could only prove it... :)

Correction: George was not born at the time of that census (1860), it was another brother so pardon my mistake. :)

This looks like the same family in 1855/60. In 55 it states Barney & Almira Thompson were born in Nova Scotia and 1860 in Massachusetts. In both census it give the occ. carpenter

Massachusetts, State Census, 1855
Name: Almira C Thompson
Gender: Female
Birth Year: abt 1832
Birth Place: Nova Scotia
Residence: New Salem, Franklin, Massachusetts, USA
Enumeration Year: 1855
Age: 23
Household number: 62
Reel Number: 10
Volume Number: 14
Household Members:

Barna Thompson 26
Almira C Thompson 23
Edward B Thompson 0
Almira A Thompson 0
Julia Thompson 0
.................................................................................


United States Census, 1860
Name: Barney Thompson
Event Type: Census
Event Year: 1860
Event Place: New Salem, Franklin, Massachusetts, United States
Gender: Male
Age: 32
Race: White
Race (Original): [Blank]
Birth Year (Estimated): 1828
Birthplace: Massachusetts
Page: 73
Affiliate Name: The U.S. National Archives and Records Administration (NARA)
Affiliate Publication Number: M653

Barney Thompson M 32 Massachusetts
Almira Thompson F 29 Massachusetts
Edward B Thompson M 8 Massachusetts
Almira A Thompson F 7 Massachusetts
Julia A Thompson F 5 Massachusetts
Daniel W Thompson M 4 Massachusetts
Josephine E Thompson F 3 Massachusetts
Ida L Thompson F 2 Massachusetts
John Thompson M 0 Massachusetts

This looks like the death of their daughter, again both parents born Nova Scotia

Massachusetts, Deaths
Name: Almira A. Thompson
Event Type: Death
Event Date: 08 Nov 1861
Event Place: Monson, Mass
Residence Place: Monson, Mass.
Gender: Female
Age: 9
Birthplace: Walpole, Mass.
Birth Year (Estimated): 1852
Father's Name: Barney Thompson
Father's Birthplace: Nova Scotia
Mother's Name: Almira Thompson
Mother's Birthplace: Nova Scotia
Reference ID: v 147 p 362 , GS Film number: 960178 , Digital Folder Number: 4220468 , Image Number: 386

*** name variation Alvina / Almira. ** again Nova Scotia

1870 United States Federal Census
Name: Alvina Thompson ****
Age in 1870: 36
Birth Year: abt 1834
Birthplace: Nova Scotia **
Home in 1870: Orange, Franklin, Massachusetts
Race: White
Gender: Female
Post Office: Northfield
Value of real estate: View image
Household Members:
Name Age
Alvina Thompson 36
Edward Thompson 18
George W Thompson 8

States second marriage on register but does not state if widow or divorced.

Massachusetts, Marriage Records, 1840-1915
Name: Elmira C Thompson
Birth Year: abt 1834
Birth Place: Nova Scotia **
Marriage Date: 13 Jul 1873
Marriage Place: Boston, Massachusetts, USA
Age at Marriage: 39
Father: Daniel Strong
Mother: Susannah Strong
Spouse: Andrews J Thompson
Spouse Birth Place: Bartlett New Hampshire
Spouse Age at Marriage: 44
Spouse Father: Miles Thompson
Spouse Mother: Mary Thompson
....................................................................................
This looks like Barney remarried

1880 United States Federal Census
Name: Barney Thompson
Age: 53
Birth Year: abt 1827
Birthplace: Nova Scotia **
Home in 1880: Danbury, Fairfield, Connecticut
Race: White
Gender: Male
Relation to Head of House: Self (Head)
Marital Status: Married
Spouse's Name: Helen Thompson
Father's Birthplace: Nova Scotia
Mother's Birthplace: Nova Scotia
Occupation: Carriage Maker

Barney Thompson 53
Helen Thompson 39

Could also go by Barnabas Thompson

United States Census, 1900
Name: Barnabas Thompson
Event Type: Census
Event Year: 1900
Event Place: Danbury city Ward 3, Fairfield, Connecticut, United States
Gender: Male
Age: 65
Marital Status: Married
Race: White
Race (Original): W
Relationship to Head of Household: Head
Relationship to Head of Household (Original): Head
Years Married: 25 ( 1875 )
Birth Date: Jul 1835
Birthplace: Canada (Eng) Nova Scotia
Marriage Year (Estimated): 1875
Immigration Year: 1856
Father's Birthplace: Canada (Eng) Nova Scotia
Mother's Birthplace: Canada (Eng) Nova Scotia

Barnabas Thompson Head M 65 Canada (Eng) Nova Scotia
Helen E Thompson Wife F 50 Massachusetts

District: 59 , Sheet Number and Letter: 7B , Household ID: 166 , Line Number: 69 , Affiliate Name: The U.S. National Archives and Records Administration (NARA) , Affiliate Publication Number: T623 , GS Film Number: 1240132 , Digital Folder Number: 004118691 , Image Number: 00960

Might be worth double checking this entry out, from LDS site

Daniel Strang
birth: 1790
Prince, Prince Edward Island, Canada
death: 13 July 1884 Searletown, Prince Edward Island, Canada

spouse:
Susanna Chappell
birth: 5 June 1798
marriage: before 1817
death: 29 April 1856

Daniel and susanna Strangs children

George Chappell Strang
birth: 14 October 1817
death: 18 May 1894

Margaret Wells Strang
birth: 3 June 1819
death: 18 March 1906

Jesse Strang
birth: 1821
death: 9 December 1881

Sarah Jane Strang
birth: May 1821
death: 2 March 1857

Eliza Ann Strang
birth: 1826
death: 1889

William C Strang
birth: about 1828
death: 1908

Almira Strang ******* Prince Edward Island, Canada ????
birth: 1828
death:

Maria Strang
birth: about 1830
death: 9 March 1880

Cyrus Daniel Strang
birth: 1832
death: 30 April 1906

Joseph Bent Strang
birth: 11 May 1836
death: 28 June 1850

John Alexander Strang
birth: 6 March 1840
death: 28 June 1863

Susannah Mary Strang
birth: 31 March 1842
death: 7 November 1855
...............................................................................

Daniel's parents
Jesse Strang
birth: 1764 White Plains, Westchester, New York, USA
death: June 1836 North Bedeque, Prince, Prince Edward Island, Canada

Sarah Wright
birth: 1769 Westchester, New York, USA
death: 1836 North Bedeque, Prince, Prince Edward Island, Canada

That is the Almira Strang I am researching. My family descends from William C. Strang. I haven't found many records from Prince Edward Island. This family moved into Ellsworth, Maine and then down to Massachusetts.

I live in Mass and I have a copy of Almira & Andrews Marriage document and her head stone says she was born 1833