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
(About Irish Civil Records, if you haven't used it before.
You'll need to click to say you're not a robot, and then enter a name & checkmark to agree to privacy rules... There's no sign up required though)
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
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
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?