Witherspoon is a reasonably common surname in the US (the 1,781st most common surname!). It comes from the Middle English words weðer, meaning ram (a male sheep) and spang, meaning a narrow strip of land. A modern and literal translation would be Ram Isthmus1. It was originally the name of a village located somewhere in the north-east of England, most likely in County Durham, where the family name Witherspoon is omnipresent2. The actual location has been lost to time, but the name lives on; strangely enough, the most famous Witherspoons have been American, particularly the American Revolutionary War big shot John Witherspoon3 and the American actress Reese Witherspoon.
The name Wetherspoon, best known as a chain of English pubs, is a common variant of Witherspoon. Wetherspoon is more common in Scotland, but it comes from the same root. Other variations still in use today include, but are not limited to, Wotherspoon, Weatherspoon, and Wetherspoon.
1. Spang, and its cousin, spong, could be used for narrow strips of land other than isthmi. While given its general location there is a good chance that weðerspang was an isthmus, it is also possible that it was a ridge or a wedge of land between to converging rivers.
2. the village of Birtley, near the towns of Washington and Sunderland, has the highest percentage of Witherspoons, but this doesn't necessarily mean much after +/-1000 years.
3. As no one has noded John Witherspoon (1723-1794) yet, I will mention here that he was born in Scotland, but came to America to help us escape the clutches of the British. He was one of the original signers of the Declaration of Independence and was president of The College of New Jersey, which eventually became what we today call Princeton University.