Name that is extremely crude to give a relative (5)
I believe the answer is:
niece
'a relative' is the definition.
(niece is a kind of relative)
'name that is extremely crude' is the wordplay.
'name' becomes 'n' (common abbreviation e.g. on forms).
'that is' becomes 'ie' ('i.e.' can be a synonym of 'that is').
'extremely' means to remove the middle letters.
'crude' with its middle removed is 'ce'.
'n'+'ie'+'ce'='NIECE'
'to give' is the link.
(Other definitions for niece that I've seen before include "member of different generation" , "A sibling's daughter" , "Preparation for curdling milk" , "Kinswoman" , "Family relation" .)