Scottish, from the port town at the mouth of the River Leith, near Edinburgh.  The name of the river is from Gaelic meaning "wet".  There is a similar word in Welsh, usually translated as "moist, damp".

Our correspondent [LD²] notes that the name occurs in Australia as both male and female names.  The name occurs a few times in the 1841 Census of Aberdeenshire but only as a male name.