Ultimately, prerogative mine to assume the nickname (7)
I believe the answer is:
epithet
'nickname' is the definition.
(thesaurus)
'ultimately prerogative mine to assume the' is the wordplay.
'ultimately' suggests the final letters.
'mine' becomes 'pit' (down the pit).
'to assume' means one lot of letters goes inside another.
The final letter of 'prerogative' is 'e'.
'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" , "Descriptive word or phrase added to a name" , "nickname" , "Descriptive adjective" , "title" .)