Start being eccentric (5)
I believe the answer is:
begin
'start' is the definition.
(synonyms)
'being eccentric' is the wordplay.
'eccentric' is an anagram indicator.
'being' anagrammed gives '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" .)