Idea unacceptable around island (6)
I believe the answer is:
notion
'idea' is the definition.
(I know that notion is a type of idea)
'unacceptable around island' is the wordplay.
'unacceptable' becomes 'not on' (something 'not on' is unacceptable or frowned upon).
'around' is an insertion indicator.
'island' becomes 'i' (geographical abbreviation).
'noton' going around 'i' is 'NOTION'.
(Other definitions for notion that I've seen before include "Whimsical idea or small sewing article" , "An idea or casual thought" , "Idea, concept" , "Vague idea or opinion" , "Idea, belief" .)