Family member has no right to show great happiness (7)
I believe the answer is:
Here is my best explanation:
'great happiness' is the definition.
(I've seen this before)
'family member has no right' is the subsidiary indication.
'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' is the link.
(Other definitions for elation that I've seen before include "Feeling of delight or pride", "Feeling of joy from toenail", "Exhilaration", "Great joy", "Great happiness and exhilaration", "Transport", "Joy", "Feeling of delight, joy", "'High spirits, exhilaration (7)'" and "It's a great feeling!".)