First I would suggest you try and find your fiancée's G-Grandmother's details in either the 1901 or 1911 census (http://www.census.nationalarchives.ie/
This will give valuable information on her year of birth, parents names and address (at least at the time of the census).
With info such as date/ place of birth and parent's names it is quite easy to order a birth certificate online from http://www.birthsdeathsmarriages.ie/
What needs to be done next is to prove the connection between your father-in-law and his grandmother - marriage certs, birth certs etc.
If you can't find her in the census - don't despair as there are still parish records. These however are not available on-line and can be a bit tricky to find unless you know exactly which church she was baptised in.
If your father-in-law is serious about getting an Irish passport I would suggest he contact the professionals such as Eneclann (http://www.eneclann.ie/
) - these are the guys called in to find the Irish ancestry of US presidents and Irish Grannies for soccer players.
So yes - if your father-in-law can prove his grandmother was born in Ireland he can claim Irish citizenship (and British also as I imagine she was born before the foundation of the Republic) but no - ship's manifests, passenger lists, immigrant info from Ellis Island etc won't do. You will need original birth certificates etc.