There is a wide spread confussion about the name Iñigo in Spain but by any chance Inigo can be English. Much earlier than Inigo Jones lived, this was a common name in the Basque country and in the heart of the castilian territory. There are several Kings in the peninsular kingdoms with that names during the obscure days of the high middle ages.
Some people relate Iñigo to the basque Eneko, and others to an antique form of Ignacio in Castile. No agreement.