Case of niece that was close relative (6)
I believe the answer is:
nephew
'relative' is the definition.
(type of relative)
'case of niece that was close' is the wordplay.
'case of' means to remove the middle letters (like emptying a case).
'that was close' becomes 'phew' (both are said after narrowly avoiding danger).
'niece' with its centre removed is 'ne'.
'ne'+'phew'='NEPHEW'
(Other definitions for nephew that I've seen before include "Family relative" , "Kinsman" , "A young male relation" , "Sister's son" , "Male relative" .)