Family boss, a Dubliner briefly stabling Arab horse (9)
I believe the answer is:
patriarch
'family boss' is the definition.
'a dubliner briefly stabling arab horse' is the wordplay.
'a dubliner' becomes 'patrick' (common Irish name).
'briefly' means to remove the last letter.
'stabling' means one lot of letters goes inside another.
'arab' becomes 'ar' (abbreviation in Chambers).
'horse' becomes 'h' ('H' can be a synonym of 'horse').
'patrick' with its final letter taken away is 'patric'.
'patric' placed around 'ar' is 'patriarc'.
'patriarc'+'h'='PATRIARCH'
(Other definitions for patriarch that I've seen before include "like Abraham was" , "Abraham or Isaac, perhaps" , "Father figure" , "tribal head" , "Male head of family or tribe" .)