Rest running after greeting quiet trendy person (7)
I believe the answer is:
hipster
'trendy person' is the definition.
(I've seen this before)
'rest running after greeting quiet' is the wordplay.
'running' is an anagram indicator.
'after' means one lot of letters go next to another.
'greeting' becomes 'hi' (hi is a kind of greeting).
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'rest' anagrammed gives 'ster'.
'hi'+'p'='hip'
'ster' after 'hip' is 'HIPSTER'.
(Other definitions for hipster that I've seen before include "Fashionable person" , "Cool dude" , "Follower of the latest fashions" , "jazz he likes" , "Bohemian" .)