The marriage of Samuel Williamson & Sarah Martin was in New Monkland in June 1850. One of their children was Martha Williamson, born in Bo'ness, West Lothian on the 15th November 1856. Sarah and Martha were living in Bo'ness in 1861 (without Samuel)
To help geographically - Merryston was a small mining community in the parish of Old Monkland (which consisted of the town of Coatbridge and some surrounding villages, including East Merryston). Both of Sarah's children mentioned on the 1851 census were born in the neighbouring parish of New Monkland, which consisted of the town of Airdrie (Robert's birthplace)and some surrounding villages, including Ballochney (Hugh's birthplace).
Note the correct spellings of Airdrie and Ballochney. I was brought up in Airdrie, so know the area very well.