(Isle of) means probably channel island. (Celtic gwy, water; gwyth, the channel.) The inhabitants used to be called Uuhtii or Gwythii, the inhabitants of the channel isle.
According to the famous Anglo-Saxon Chronicle, the island is so called from Wihtgar, great grandson of King Cerdic, who conquered it. All eponymic names- that is, names of persons, like the names of places, are more fit for fable than history: as Cissa, to account for Cissanceaster (Chichester); Horsa to account for Horsted; Hengist to account for Hengistbury; Brutus to account for Britain; and so on.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Wight from Infoplease: