According to Tunisian olive museum (Zaitounah Museum) the olive tree came to Palestine from Armenia after which it spread to the Mediterranean and North Africa. It is explained in the following news article from 2004.
 
Zaitounah Museum new addition to Tunisian heritage
 
The Zaitounah (olive) Museum which was opened recently in the eastern city of Sousse is considered a new addition to the Tunisian cultural heritage due to the pivotal role the olive tree plays in the lives of Tunisians and the population of the Medieterranean region. […] Gadira said that it also aims at preserving the heritage of this “blessed” tree for future generations, noting that this tree remained steadfast before the various civilizations which ruled Tunisia over some 3000 years like the Berbers, Phoenicians, Byzantines, Arabs, Spaniards and Turks. He said that the most ancient documented sources available report that the olive tree was brought into Palestine…