One seeking legal redress for everyday quarrel (9)
I believe the answer is:
plaintiff
'one seeking legal redress' is the definition.
'everyday quarrel' is the wordplay.
'everyday' becomes 'plain'.
'quarrel' becomes 'tiff' (tiff is a kind of quarrel).
'plain'+'tiff'='PLAINTIFF'
'for' acts as a link.
(Other definitions for plaintiff that I've seen before include "Bringer of lawsuit" , "One bringing legal case" , "The one taking the court action" , "accusing person in suit" , "The litigant making the accusation" .)