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FaNUK is a project set up at the The Bristol Centre for Linguistics at University of the West of England. Their aim is to consider the origins of surnames from the etymological roots, ie from how words evolved. This may have little to do with any place-names or relationship-based ideas (eg ‘son of’).
Their thoughts have been presented in a dictionary, to which I have contributed early references and thoughts on the potential origins. The entry for GOTT and GOTTS is shown in this pdf document.
They have relied on the discussion of Gott which is prevalent in Yorkshire, but also occurs in Norfolk and Lincolnshire.
Personally, my research shows the names GOTT and GOTTES appear mixed in the same document, eg a lease in 1633 has GOTT twice and GOTTS twice, and a will of 1529 has the spelling GOTTS of the testator at the top of the will, and GOOTIS in the margin for indexing, so the general level of education and grammar ends up with the spellings being used interchangeably. My feelings are that people with either name around the Wash, ie Lincolnshire and Norfolk may have the same origins, whilst GOTT in Yorkshire will be a separate source, though ultimately the names could all be derived from earlier Viking or Anglo-Saxon invasions. It does not follow, however, that the families are linked back at that time, just that the name has derived from their locations in 1200-1300’s from a use of a common language.