The first known person named "Odding" was the father of William Odding named John Odding. He and his wife lived in Braintree, Essex, England. His son William brought the name further and it ended up in the U.S. with her son's widow Margaret and her daughter Sarah Odding who emigrated in 1630.
The name has since spread to many countries:
Norway: Hakon Odding received by the Justice Department's licensing 17 August 1927, permission to assume that Odding family name instead of Eilertsen. This message is the profile of all those who have the name in Norway.