Great joy when family member doesn't start (7)
I believe the answer is:
elation
'great joy' is the definition.
(I've seen this before)
'family member doesn't start' is the wordplay.
'family member' becomes 'relation' (a relation or relative).
'doesn't start' means to remove the first letter.
'relation' with its initial letter removed is 'ELATION'.
'when' acts as a link.
(Other definitions for elation that I've seen before include "Bliss" , "Euphoria of toenail" , "Feeling of delight, joy" , "There's great joy" , "Ecstasy -- toenail (anag)" .)