This is largely a myth. There were many non-Jews in Israel during Second Temple times, including a very large Greek community, and huge Samaritan community.
Some Jews in ancient Judea certainly converted to Christianity. But the Jewish Church basically got annihilated during the revolts against Rome. Today's Israeli Arab Christian community largely descends from a combination of Greeks, Crusaders, Samaritans, and pious pilgrims throughout the centuries. Jews are likely a pretty small portion of the community's ancestry. If it makes you feel better, Samaritans are largely Israelite (though not Jewish). And those who weren't killed by the Byzantines largely converted to Christianity and, later on, Islam.
It should also be pointed out that the entire Jewish community was not immediately exiled from Judea, as apposed to Jerusalem. After the destruction of the Temple their remained a Jewish community in the countryside. This community dwindled for a number of reasons. There had already been a large and voluntary exodus of those who sought a better life in other parts of the empire. Thus there were large Jewish communities in all the major cities of the empire before the rebellion. This would continue as pressures increased after the rebellion. Of those that remained, some would convert to Christianity and others would intermarry with the non Jewish population. All of this happened before the Arab invasion. True, the Christian population was composed of non-Jewish elements as well. The honest answer is that there is no way to know what percentage of the Christian Palestinian population is Jewish but it cannot be denied that it is one element.
In the end, however, I do not think that it is important. As a Christian, I follow St. Paul who said that there is no longer a distinction between Jew and Greek. In Jesus Christ the gentile Christians have been grafted onto the root of Judaism and are joint heirs with them. I do not expect Jews to accept this but at the same time Jews should not expect Christians to believe that Jews continue to have a separate covenant with God that excludes the Christians. As long as we respect on another and treat each other with dignity that is enough.