Mythical hero being cool in rising water (8)
I believe the answer is:
achilles
'mythical hero' is the definition.
'cool in rising water' is the wordplay.
'cool' becomes 'chill' ('chill' can be a synonym of 'cool').
'in' is an insertion indicator.
'rising' is a reversal indicator (going up in a down clue).
'water' becomes 'sea' (sea is a kind of water).
'sea' written backwards gives 'aes'.
'chill' put inside 'aes' is 'ACHILLES'.
'being' is the link.
(Other definitions for achilles that I've seen before include "Weak-heel warrior" , "Ancient Greek with weak heel" , "Greek hero of the Trojan War" , "Great Greek with weak heel" , "Cause of Hector's death" .)