Most certainly is an area name. Named after a standing stone that was once on the spot known as Cross-Stobbs Cross. The general area had several standing stones: The Arthurlie, Capelrig, Steed-Stane Cross, Cross-stobs, and a little further along, the Neil’s-stone (a la Neilston).
The Cross Stobbs pub didn't invent the name for itself. It named itself after the area name.