Eccentric included an unfounded rumour (6)
I believe the answer is:
canard
'unfounded rumour' is the definition.
(I've seen this before)
'eccentric included an' is the wordplay.
'eccentric' becomes 'card' (I've seen this before).
'included' means one lot of letters goes inside another.
'card' going around 'an' is 'CANARD'.
(Other definitions for canard that I've seen before include "A French kind of false rumour" , "A false rumour from France? (6)" , "False rumour, originally from France" , "Deliberately misleading fabrication" , "rumour that's false" .)