Record store has name of Greek character (7)
I believe the answer is:
epsilon
'greek character' is the definition.
(letter from the Greek alphabet)
'record store has name' is the wordplay.
'record' becomes 'EP' (abbreviation for extended play record).
'store' becomes 'silo' (a store of crops).
'has' says to put letters next to each other.
'name' becomes 'n' (common abbreviation e.g. on forms).
'ep'+'silo'+'n'='EPSILON'
'of' is the link.
(Other definitions for epsilon that I've seen before include "Foreign character" , "foreign letter" , "small number mispronounced, almost completely the wrong way round" , "Fifth letter of the Greek alphabet" , "Greek character" .)