Hi Cindy. I've found a Mary Burns born in 1868 with a father named Charles. The birth date is different than the one I see on FS, but those kinds of things can get mixed up for orphans (my Great Grandpa had a similar problem - they thought he was 3 when he was taken in, but he was really 5).
Of course, it's quite possible there are several Mary/Charles relationships.
Here's the certificate
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1868/03444/2264517.pdf
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I've also found an Agnes Byrne with a fathers name that looks like Chs Byrne (I believe that's an abbreviation for Charles) on 2 April 1867 in Rathmichael, Rathdown, Co Dublin, Ireland
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1867/03487/2289254.pdf
It also looks like this family (Charles Byrne and Catherine Cooke) had another child Mary Anne Byrne on 7 April 1869, also in Dublin. Ancestry link https://search.ancestry.com/cgi-bin/sse.dll?indiv=1&dbid=2533&h=151378765
And these other children:
Christopher Byrne, 15 May 1872
Charles Byrne 25 May 1878
And a death record for Charles Byrne in 1876 (age 29) which would line up with being an orphan at 10 years old
https://search.ancestry.com/cgi-bin/sse.dll?indiv=1&dbid=2534&h=1680284
And there are some 'prison register' records that might be worth looking at for Catherine Byrne around that time.
Just theories at this point, but maybe there's a lead in here somewhere?
Cheers,
Loralee