Thank you again for the reference. It had been a while since I had looked, but I see that Mary has unfortunately still not turned up there.
At the genealogy center in Sligo, they found a Mary born in 1863 in Ballyglass, not 1868, whose parents were Michael and Mary. Michael's wife's maiden name is listed as Henry.
At the GRO in Sligo they found a Robert born to a Michael and Mary (Marren) Armstrong. After I returned from Ireland I discovered that Patrick Conboy's year of birth was 3 years earlier than maintained by my relatives and than that which appears on his gravestone in Rockland County, New York. The day of birth was the same. So, perhaps Mary Josephine was born earlier (or later) as well.
In Irish death records there are four Michael Armstrongs in Tobercurry that theoretically could have been her father:
Oct-Dec 1903 aged 68 (b. 1835),
Oct - Dec 1904 aged 84 (b. 1820),
Jan - Mar 1906 aged 75 (b. 1831),
Oct - Dec 1929 aged 89 (b. 1840).
I know that a Michael and Mary are buried at Ballyglass, but the gravestone was being restored the only time I was able to visit the area.
I have sighted the following possible siblings in Tobercurry records: Thomas (b. 1 Jul 1864), John (b. 1864), Margaret (b. 15 Aug 1866), John (b. 1866), John (b. 1867), Serah (b. 29 Apr 69), Thomas (b. 1869), Honor (b. 5 Jan 1874), Thomas (b. 1875), Michael (b. Apr - Jun 1879). What family members were always conform about was that she was the only girl with 6 brothers who emigrated to England. The only name of the brothers anyone could remember was Jim!!! No one mentioned any deaths of siblings at young ages, but it cannot be ruled out.
We know that Mary Josephine's husband immigrated to NY on 23 Apr 1888, coming with the SS Richmond in the year of the great blizzard. We have not found Mary Josephine's entry date and only know they were married 4 years after his immigration.
Beginning in 1900 the US census information often contains dates of naturalization and later the year of immigration. May be some clue there.