Idea that’s unacceptable about island (6)
I believe the answer is:
notion
'idea' is the definition.
(I know that notion is a type of idea)
'unacceptable about island' is the wordplay.
'unacceptable' becomes 'not on' (something 'not on' is unacceptable or frowned upon).
'about' means one lot of letters goes inside another.
'island' becomes 'i' (geographical abbreviation).
'noton' placed around 'i' is 'NOTION'.
'that's' acts as a link.
(Other definitions for notion that I've seen before include "Caprice" , "clue" , "A vague and possibly incorrect opinion" , "Vague or fanciful idea" , "Idea, belief" .)