The names Goatham and Gotham seem to have originated in England, although there are probably now more Gothams in the US descended from English Gothams or Goathams, and some have other origins.
It is almost certainly a locative surname, i.e. from a location named Gotham. Although most well known is the village of Gotham in Nottinghamshire, there are a number of smaller Gothams to be found in different parts of the country and it seems to be from one or more of these that the forebears of those of us with the surname today descend.
There seem to be 3, possibly 4, distinct groups of Goathams / Gothams / Gottams who may or may not share a common ancestor.
At one time the family(ies) were most numerous in Devon, where all three of the above surname variants occurred, as well as a range of other spellings, although Gotham was most common and it is that surname that all descendants of the Devon group seem to have ended up with. All Gothams in Devon in the last century or so are descended from one William George GOTHAM ancestor, born about 1818 in Teignmouth; other lines became died or daughtered out, and/or left Devon.
In Kent as would be expected a variety of spellings have been used in the past; both Gotham and Goatham were commonly used and the name has finished up as Goatham. After centuries of stuggling to survive, probably due at least in part to living in unhealthy areas where malaria was rife, the Goathams multiplied quite rapidly in the nineteenth century.
Shropshire / Staffordshire
A small number of Gottams / Gothams lived in parishes along or close to the northern half of the Shropshire / Staffordshire border, at least from 1616/7 when the marriage of a John Gottam took place in Gnosall. In the rapid expansion of Stoke-on-Trent as an urban area some moved the short distance to that area, but so did a Devon Gotham; I have not yet done more than glance at this Midlands branch so I don’t know if it expanded, contracted or emigrated and so those in this area now may be from either group.
A little further north a couple of Gothams were born in Clitheroe in the mid-C19th, John Francis in about 1842 and Margaret in 1861. John Francis’ grandson, Francis Ernest, kept the line going until his death in 1974, when the family daughtered out.
I don’t know how they were related to any other Gothams. Whilst most ‘strays’ turn up in London, a James of Liverpool married in nearby Aughton in 1703. Maybe they were descended from him. Although in the same county, Clitheroe and Liverpool are not close (about 50 miles apart) but Clitheroe is closer to Liverpool than to any other know Gothams.