There is certainly a possibility but Halliday wasn't that uncommon a name in Lanarkshire. If your grandfather Halliday was from the Wishaw area (consider too place names such as Cambusnethan, the Mondklands, Carluke ), the likelyhood increases!
My ancestor John Halliday, though born at Crawford around 1804, lived, worked as a coal miner and died (1853) in the vicinity of Wishaw - so did the next generation. Working in the mines was what the men in this family did, for the most part.
John's parents, Robert Halliday and Margaret Williamson, appeared to have ended their days in there as well. Another Halliday family who lived side by side with mine, that of Archibald Halliday and Marion Penman. I believe these two families are closely connected but I can't yet be sure.
The theory I am going with at the moment is that Archibald and Robert were brothers born at Killin, Perth, Scotland, children of THOMAS HALIDAY and ANNE CAMPBELL. But that remains but a theory!
cheers,
ard