Hello Tony, I wondered if this is what you are looking for?
There's a death on ScotlandsPeople for a Catherine Boyle, age 0, in 1956, and the registration district is Aidrie / New Monkland which would include Plains. There's nothing for a Kathleen with a K.
It might be worth looking at the certificate on line, because that will give you her parents' names and you'll know if they are your grandparents or not.
To view it, go to Scotland's People by clicking the link sancti gave. You need to register to search, but that costs nothing. Once you have you'll need to purchase credits, but the smallest purchase of 30 credits will cover what you need for this, with plenty over.
You want to search 'Deaths' (3rd from the top on the left hand menu) and put in Catherine Boyle, 1956, then select Lanark as the county and Aidrie as the town. It should come up as '1 record, cost 1 credit to view'. Agree and you will get what I just saw, showing a GROS (reference number) of 651/00 0257. Click on 'View (5 credits)' and it will show you the certificate