Hi Bill,
I do believe that the records you have found for Mary and Maria Brannigan are correct in that they are one and the same, Brannigan is not a common surname in Killybegs as it it may as be in other Parishes within Co.Donegal in the 1850's up to the 1900's. You have done well to find Records for your 2nd Great Grandmother from back in the 1840's. I have across many of my own family that were born in the mid to late 1800's that emigrated to the US. When I found their US Death Certificate very few of them, if any at all would suit the dates of their Birth or Baptism recorded in the local Church Records, and I know for sure that I do have the correct Records from both sides of the Atlantic because shared Family DNA has confirmed this for me.
Unfortunately, the bad news I have for you is that Birth Records for Catholic Families in Donegal only started to be recorded in the 1840's, and so a Birth Record for William Brannigan or a Jane McGonagle (a fine Donegal name) will not be available to be found on any search. The Records and Years of Register for Donegal are as such:
http://donegal.rootsireland.ie/generic.php?filename=centres/donegal/sources.tpl
If you are interested in still exploring this family from Killybegs Upper further, the best advice I can give you would be similar to Janes above, putting together an Ancestor profile for Mary Brannigan. This is something which is worthwhile and when done right it makes it so much easier to connect to local Brannigan's still living in the Killybegs area. I have had some success with this in the past, and could keep you right if you need.
The family I would be interested in reaching out to is a Thomas Branagan Family who had a House in Carricknamoghil (Killybegs, Donegal) in the 1901 and 1911Census. As I said before there is no other Brannigans living nearby within the Parish and the fact that they come from the exact location of William Brannigan was in 1850's means they were closely related, possibly even brothers.
http://www.census.nationalarchives.ie/pages/1901/Donegal/Killybegs/Carricknamoghil/
http://www.census.nationalarchives.ie/pages/1911/Donegal/Killybegs/Carricknamoghil/
If you are interested
Here would be a few examples of profiles that might give a few ideas for Mary Examples of Ancestor Profiles
Documenting who her parents were and her family life going forward would be interesting for locals from Killybegs area
Because it is so long ago chances are you wouldn’t have a photo of Mary but if you look through the Mulroy Drive facebook site you will find a few photos from Killybegs to place up instead HERE
When sharing your ancestor I would recommend that you post it in the Upper Killybegs and 4 or 5 surrounding parishes also. A list of Surrounding Parishes can be found here HERE
I am curious to hear what do you think, it is in your well capable hands now!
Best Regards
Seamus