Family member has no right to show great happiness (7)
I believe the answer is:
elation
'great happiness' is the definition.
(I've seen this before)
'family member has no right' is the wordplay.
'family member' becomes 'relation' (I've seen this before).
'has no' is a deletion indicator.
'right' becomes 'r' (common abbreviation).
'relation' with 'r' taken out is 'ELATION'.
'to show' acts as a link.
(Other definitions for elation that I've seen before include "Great happiness and exhilaration" , "It's a great feeling!" , "Delight, thrill" , "State of being delighted or proud" , "Transport" .)