Point to mine having the nickname (7)
I believe the answer is:
epithet
'nickname' is the definition.
(thesaurus)
'point to mine having the' is the wordplay.
'point' becomes 'e' (E is an example).
'to mine' becomes 'pit' (down the pit).
'having' indicates putting letters inside.
'pit' going around 'the' is 'pithet'.
'e'+'pithet'='EPITHET'
(Other definitions for epithet that I've seen before include "Sobriquet" , "qualifier" , "Fitting description" , "nickname" , "Tag" .)