Start being eccentric (5)
I believe the answer is:
begin
'start' is the definition.
(I know that start is a kind of begin)
'being eccentric' is the wordplay.
'eccentric' indicates an anagram.
'being' is an anagram of 'BEGIN'.
(Other definitions for begin that I've seen before include "Get cracking" , "Found" , "Israeli prime minister who shared the Nobel Peace Prize with President Sadat" , "Make a start" , "Israeli prime minister and Nobel Peace Prize winner, d. 1992" .)