Its Latin name was Vectis and 'v' was sometimes pronounced 'w' in that language. Eventually it transferred into English as 'Wiht' originally, from where it became 'Wight.
What can I say? 'Wight' is an anglicised version of 'Vectis' pronounced 'Wectis' and shortened by having the ending removed. The 'Isle of Dogs' is never called just 'Dogs' as far as I know either!
The Isle of Man is similar - it's never known just as Man. It's just a language thing - some islands were called by non-island/isle names (like Trinidad) and others were tagged with Isle of, or Island (you must remember Fantasy Island?).