The name Gough was the subject of an article in the Journal of One Name Studies, Vol. 13 Issues 6, Apr-Jun 2019.
The article is by Phillip G. Goff, pp 20-23, and is part 1 only. He starts by showing that the surnames Gough and its variants are not related to the surnames Goff and its variants; Gough being from the counties that border Wales and the name Goff being from the south-east counties of England and into the Midlands.
So, turning to the origins of 'Gough' he shows that the earlier and common spelling back in the 1600's and before was Gogh or even Goch and derives from a nickname derived from the Welsh coch or goch meaning 'red-headed'. Apparently there is some DNA evidence that the gene MC1R variants are more common in the group.
By the 1800s the spelling of Gough become more prevalent and the Gough families moved around the country as ours did!