Possibly named after the Scottish royal house of Canmore. The first of the line was Malcolm III (c. 1031-93). He acquired the nickname Canmore, from Gaelic Ceann Mór, probably meaning 'big chief', after coming to the throne. In theory, Ceann Mór could be a place-name but there is no place called Canmore in Scotland. The nearest thing is a village called Kenmore at the head of Loch Tay in Perthshire, which is also derived from Ceann Mór. Therefore, it seems most likely that Canmore Street is derived from the royal clan.
There are several other streets with Scottish names in the Shankill district, such as Argyle Street, Aberdeen Street, Dundee Street and Cupar Way.