Ireland's documents are few and far between, both parish and government.
earliest census substitute 1823-1837 :www.genealogy.nationalarchives.ie
There are 178 Keeffe's in Cork
next census substitute 1851 Cork Griffith Valuation
Lysaght, 22 in Cork, 5 Fermoy Parish and 1 Fermoy Parish and Fermoy Townland only two first names Mary x 4 and John.
This could be 1 Mary or 4 or any combination
There are 1,057 Keefe's in the Griffith Valuation and also David was quite a rare name at the time there are 39 David Keefe's in Cork.
There are also 24 Arthur's but only 1 in the Parish of Fermoy, Townland and Town of Fermoy in Fermoy Poor Law Union
Carroll, no Michael but 4 Carroll's in Fermoy Parish, names are Richard and William (so perhaps William was named after
Michael Carroll's father) perhaps they hadn't married yet?
Only I wasn't able to get the details of the land for either as Keefe or O'Keefe on www.askaboutireland.ie
?? very strange as they are on my program of Griffith Valuation but not on www.failteromhat.com
Lysaght seems to work fine on both resources.
There are baptisms on line for Fermoy RC 1827-1922 and marriages 1828-1922, Church of Ireland 1822-1903 for bothwww.rootsireland.ie
(pay not for profit Foundation)
I'm surprised as you have both first and last names of the parents you can't find William as a sibling?
(free) see church records, I believe this is only for the South West of Ireland but worth a look.
William, was he the youngest? maybe he was born on the voyage? and why there is no record in either Ireland or
There are few death records in parish's both RC and Cof I, although there are some. Too early for the civil registrations of 1864.
If Mary remarried, her husband should of course have died, but sometimes the Irish because of no divorce just left and remarried.
Some early gravestone records on www.from-ireland.net
might be worth a try, I tried O'Keefe - one so you can try Keefe.
Other records that might be useful for deaths arewww.ancestorsatrest.com
+ Ireland + Corkwww.interment.net
+ Ireland + Corkwww.igp-web.com
+ Cork had parish records and cemetary's haven't checked it lately.
Good luck on your journey.