Point to mine having the moniker (7)
I believe the answer is:
'moniker' is the definition.
'point to mine having the' is the wordplay.
'point' becomes 'e' (E is an example).
'to mine' becomes 'pit' (down the pit).
'having' is an insertion indicator.
'pit' going around 'the' is 'pithet'.
(Other definitions for epithet that I've seen before include "qualifier", "Descriptive work or phrase", "Word describing", "Descriptive term used to express a real quality", "Descriptive word or phrase added to a name".)