1590s, "sanctimonious person, religious hypocrite," from French bigot (12c.), which is of unknown origin. The sense was extended 1680s to other than religious opinions.
The earliest French use of the word is as the name of a people apparently in southern Gaul, which led to the theory, now considered doubtful on phonetic grounds, that the word comes from Visigothus. The typical use in Old French seems to have been as a derogatory nickname for Normans, leading to another theory (not universally accepted) that traces it to the Normans' (alleged) frequent use of the Germanic oath bi God. OED dismisses in a three-exclamation-mark fury one fanciful version of the "by god" theory as "absurdly incongruous with facts." At the end, not much is left standing except Spanish bigote "mustache," which also has been proposed as the origin of the word, but not explained, so the chief virtue of that theory is the lack of evidence against it.
In support of the "by God" theory the surnames Bigott, Bygott are attested in Normandy and in England from the 11c., and French name-etymology sources (such as Dauzat) explain it as a derogatory name applied by the French to the Normans and representing "by god." The English were known as goddamns 200 years later in Joan of Arc's France, and during World War I Americans serving in France were said to be known as les sommobiches (see son of a bitch) for their characteristic oaths.
But the sense development in bigot would be difficult to explain. According to Donkin, the modern meaning first appears in French in 16c. This and the earliest English sense, "religious hypocrite," especially a female one, might have been influenced by or confused with beguine (q.v.) and the words that cluster around it.