If you have a look at the askaboutireland website you'll see that Griffiths Valuation, with a print year of 1852 for the local area, has quite a few people living in what is referenced as:
'STRAND ROAD OR QUAY LANE (OFF GREEN LANE)'
... which according to the maps is in the immediate vicinity of Strand Street today.
Edited to add: an 1864 newspaper carried a death notice for 'Ellen, the beloved wife of Mr Eugene Moriarty', the death was 'at her residence, the Strand, Dingle'. Griffiths Valuation has only two entries in Dingle for Eugene Moriarty, both are in 'STRAND ROAD OR QUAY LANE (OFF GREEN LANE)'.