Hi Maria, I found this old post while doing some speculative trawling for people connected to Airdriehill Square. My great-grandfather was Thomas O'Brien. He originally came from Bonmahon, Co. Waterford and was married to Mary Crotty. They moved to Scotland early 1870's and lived at Airdriehill Square, Whiterigg. As far as I know they had 3 children born in Co.Waterford, Richard Bn.1868, James Bn.1870 and Bridget Bn.1873. My grandfather, Thomas O'Brien, was born later at Whiterigg in 1882.
This is where I noticed a slight error in your post, which you may well have remedied by now.
It was my grandfather's older brother, Richard O'Brien, who married Marie McDonagh on 15/7/1891 at St. Margarets RC Church, Airdrie, at which time she lived at 18 Airdriehill Square and he lived at No.20 (viewed the marriage registration on scotlandspeople website)
My grandfather, Thomas O'Brien married Catherine Lee in 1913 at St.David's RC Church, Meadowohead, at which time they also both lived at Airdriehill Square.
My great-grandfather, Thomas O'Brien, died at Whiterigg in 1895 aged 50yrs and his wife Mary (Crotty) died, aged 75, at Whiterigg in 1922.
My father, also Thomas O'Brien, was born and brought up in Airdriehill Square, Whiterigg where he lived until the mid 1930's, when the village was more or less abandoned and people moved to the nearby villages of Plains and Caldercruix. Would be pleased to hear if you knew anymore about this side of the family.