Point to mine having the descriptive name (7)
I believe the answer is:
epithet
'descriptive name' 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' indicates putting letters inside.
'pit' placed around 'the' is 'pithet'.
'e'+'pithet'='EPITHET'
(Other definitions for epithet that I've seen before include "The 23: an -- for Richard I" , "nickname" , "qualifier" , "Word describing" , "Sobriquet" .)