What I presume must be Francis’s death certificate was registered in Londonderry on 14th July 1969. He was aged 66 (so perhaps born c 1903). You can view the original certificate on-line on the GRONI website, using the “search registrations” option:
https://geni.nidirect.gov.uk
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate. The death certificate should give you his marital status and usual address which may help narrow the search. (If you post that information here, I'll see if I can find anything more. A marriage perhaps, and it may help find the family in the 1901 & 1911 censuses).
I found a newspaper report where Francis’s doctor said he was born in May 1908. However I cannot find a birth that matches that so it may not be quite correct. (A lot of ages were guesses).
Looking at the 1911 census there was a Francis of about the right age in that general area, though without more information, I can’t say it is definitely your family:
http://www.census.nationalarchives.ie/pages/1911/Londonderry/Glenshane/Cluntygeeragh/595768/
Cluntygeeragh is about 5 miles from Dungiven. I looked to see if your Francis left a will, but he does not appear to have done.