Pat,
References to the Rodgers family of Lurganconary on this site:
http://freepages.genealogy.rootsweb.ancestry.com/%7Erosdavies/SURNAMES/R/Rodgers.htm
The tithe applotment records show Richard Rodgers in the townland in 1830.
Kilkeel RC parish records only start in 1839 so you may struggle to find anything on the family prior to that. Death registration didn’t start till 1864 so unless there is a gravestone, you may not find any record of Richard’s death.
Griffiths Valuation for 1863 shows Alice Rogers on the only Rogers property in Lurganconary (plot 4 which was a 3 acre farm). Normally when a woman is listed in Griffiths she is a widow. So that’s a hint that it could be Richard’s widow.
The revaluation records show Alice’s name against the property until 1885 when she was replaced by Richard Rogers. That suggests she may have died. Richard’s name remains against the property till 1914, when he probably died. Here he is in 1901. He was evidently born c 1840 so you might find his baptism record which should contain his parents names.
http://www.census.nationalarchives.ie/pages/1901/Down/Greencastle/Lurganconary/1235963/
And his marriage may reveal his father’s name. (Which I would guess was Richard).