This road is likely named after the landscape. “Du” or “Dun” often refers to a hill and ‘more’ is likely derived from ‘moor;’  A “hay” refers to a fenced-in area of land or enclosure to contain livestock. Dumore Hay Lane is a continuation of “The Moor”, so was probably moorland that was enclosed at some point, with the land to the north certainly raised compared to the neighbouring land.