Rum delivery? That's eccentric (7)
I believe the answer is:
oddball
'eccentric' is the definition.
('oddball' can be a synonym of 'eccentric')
'rum delivery?' is the wordplay.
'rum' becomes 'odd' (I've seen this before).
'delivery?' becomes 'ball' (ball is a kind of delivery).
'odd'+'ball'='ODDBALL'
'that's' is the link.
(Other definitions for oddball that I've seen before include "Eccentric person like uneven sphere" , "Weird" , "Strange, bizarre" , "Crank" , "Weirdo" .)