Short person from Athens can start to give time of day (8)
I believe the answer is:
greeting
'time of day' is the definition.
('good time of day to you' is an archaic greeting)
'short person from athens can start to give' is the wordplay.
'short' means to remove the last letter.
'person from athens' becomes 'greek' (I've seen this before).
'can' becomes 'tin' (synonyms).
'start to' suggests taking the first letters.
The first letter of 'give' is 'g'.
'greek' with its last letter taken away is 'gree'.
'gree'+'tin'+'g'='GREETING'
(Other definitions for greeting that I've seen before include "Welcoming" , "causes weeping for some" , "Words of salutation" , "Scottish weeping or 16 Across" , "Word of welcome" .)