My gran was a Bourhill and there was a large family of them in Motherwell.
About four years ago I met a family of Bourhills from Midlothian and they had twin boys. I clocked them right away. Same dark eyes and hair as me.
My late Mum told me the name was Hugenout from France. French Protestants exiled because of their faith!! I was told they became lacemakers in the Darvel area of Ayrshire.