Exclusive party in extremity adopting new name (3,5)
I believe the answer is:
hen night
'exclusive party' is the definition.
(party exclusively for women)
'extremity adopting new name' is the wordplay.
'extremity' becomes 'height'.
'adopting' is an insertion indicator (inserted letters are taken on or adopted).
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'name' becomes 'n' (common abbreviation e.g. on forms).
'n'+'n'='nn'
'height' enclosing 'nn' is 'HEN NIGHT'.
'in' is the link.
(Other definitions for hen night that I've seen before include "Party for women only" , "prematch party" , "then may the ladies rejoice" , "Bride-to-be's party" , "Bridal celebration" .)