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As you may or may not know, before 1864 records were kept in the form of church and parish records and all the surviving records have been sorted into county-based genealogical centres. For County Donegal there is Donegal Ancestry
and also a website Donegal Genealogy Resources
that might be worth a look. The first place may well be the actual church on Arranmore for the birth records at least.
In 1864 civil registration was introduced in Ireland, which is when we started seeing the formal registration of births, marriages, adoptions, and deaths. These records have been kept at the General Register Office of Ireland
(GRO), to this day. This would be handy for marriage and death records.Find a grave
has a Cornelius Bonner listed on their page with some details and an image of the headstones, is this your grandfather?
We would love to hear more, hopefully, you can retrieve the information you had.