Point to mine having the sobriquet (7)
I believe the answer is:
Here is my best explanation:
'sobriquet' is the definition.
(I've seen this before)
'point to mine having the' is the subsidiary indication.
'point' becomes 'e' (E is an example).
'to mine' becomes 'pit' (down the pit).
'having' indicates putting letters inside.
'pit' enclosing 'the' is 'pithet'.
(Other definitions for epithet that I've seen before include "What characterises", "Phrase expressing some quality of a person or thing", "Descriptive term used to express a real quality", "Appellation", "Description", "Sobriquet", "A descriptive word or phrase", "Descriptive adjective", "Descriptive work or phrase" and "qualifier".)