Eponymous hero’s extremely offbeat greeting (7)
I believe the answer is:
othello
'eponymous hero's' is the definition.
(I've seen this before)
'extremely offbeat greeting' is the wordplay.
'extremely' means to remove the middle letters.
'greeting' becomes 'hello' (hello is a kind of greeting).
'offbeat' with its centre taken out is 'ot'.
'ot'+'hello'='OTHELLO'
(Other definitions for othello that I've seen before include "Shakespearean play" , "Jealous character" , "As part" , "Will's hero" , "game with counters" .)