Hi Martin
The Irish government has put the civil birth, marriage and death records (not just the indexes) online for free.
https://www.irishgenealogy.ie/en/
Go to Civil Records. You have to declare your name to search the records, but it is free to do so. Putting in William Stynes as a search term, I found a William who died in 1955 aged 47 in the village of Sleaty in the Abbeyleix Registration District who was a farmer, informant was Bridget his widow.
Sleaty is on the outskirts of the town of Carlow in County Carlow, but the county line between County Carlow and County Laois has separated the two.
There is a marriage of William Stynes, a farmer to a Bridger Foy of "Sleaty, Carlow" on 20 June 1934 in the Catholic church of Graiguecullen, Carlow.
Unfortunately, the 100 year closure rule on public display of birth records would mean that your mother born in 1936 won't be searchable, but these two dates of the marriage in 1934 and the death of a possible father in 1955, would fit with the information you already have.
Jo