Point to mine having the sobriquet (7)
I believe the answer is:
epithet
'sobriquet' is the definition.
(I've seen this in another clue)
'point to mine having the' is the wordplay.
'point' becomes 'E' (compass point).
'to mine' becomes 'pit' (down the pit).
'having' is an insertion indicator.
'pit' placed around 'the' is 'pithet'.
'e'+'pithet'='EPITHET'
(Other definitions for epithet that I've seen before include "qualifier" , "Phrase expressing some quality of a person or thing" , "Descriptive term used to express a real quality" , "Descriptive adjective" , "Fitting description" .)