Hi Mary Jo . . .
Welcome to I.R.O. . . .
Because our regular (more professional).. researchers .. have passed over your query . .
I expect . . they had a quick look . . a brief search . . and decided . . given your info. .. they couldn't help . .
So . . I had a look . . mainly because I have a Son-in-Law . . named Lynskey .. in Galway County.
I hasten to add that . . I am far removed from Co.Mayo . . hence . . no local insights ....
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First . . I searched ..." Tithes Applotments " 1823 -1837 . . a Tax Roll from that time . . ( photo below )
This is a list of all the Lynskeys ... smallholders . . for Co.Mayo .. 10 (only) in total . . with Townslands.
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?…
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Because Catherine & Ellen have later Emigrated to the U.S. I then looked for a Mary . . in. .
the 1901 Census .. and found . . (see below ) ....
This Mary ...lists herself ...as " Lynskey " ???.... having a Daughter who married a Brannan . .
( this Mary does not appear in the 1911 Census .. presumed passed )
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Next . . " Griffiths Valuation " 1847 - 1864 . . . again a Property Tax roll . . .
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam…
Here are 12 Lynskeys . . 2 (two) of whom are Michaels . . giving Townslands . . .
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None of the Above . . leads to any definite info. re the Family you search for . .
But . . lots of clues . . as to te where and whats . . .
A further help site is . .
https://www.mayo.ie/library/local-history/geneaology
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So Mary Jo . . Best I could do . . .
Good Luck with your continued searching . . .
Cheers . . Eamonn.