Just some thoughts to show that whether he is Jew, Muslim, Arab or Hebrew is actually irrelevent. I'd be interested in everyone's thoughts on this but...
God gave Abraham a promise that had many parts to it, but which was actually depended on him losing his family and his homeland in obedience to the one God who had created all things.
1. God changed his name from Abram to Abraham.. Abram means exalted father and Abraham means Father of many Nations. (Genesis 17:5)
2. Gods call seems to have come to Abraham twice.. once in Ur of Chaldees and once in Haran. the first time with his father alive and the second when his father died. What is very true is that it was a call to leave his own land and God would show him another country.
3. The promise was a blessing for obedience, the word bless occurs in Genesis 12:2-3 5 times. The blessing that God gave to Abraham that He would be blessed would spill over to ALL nations and ALL mankind.
A land, a posterity and a blessing are each discussed in the following chapters.
God gave the Land that was in the fertile Jordan valled to be his forever. "Lift up your eyes and look from the place where you are, northward and southward and eastward and westward: For all the land which you see now I will give to you and to your descendents forever. Gen 13:14-15
Second the posterity. God told Abraham to look at the stars and told him who at that time had no children, that his descendents would be as numerous as the stars. (In Hebrews we read that on the basis of him believing this almost unbelievable promise, he found salvation.. Hebrews 11)
Thirdly, he was blessed to be a blessing. God told Abraham he would confirm a covenant with him .. a covenant to establish his descendents forever as His (God's people). And he gave them circumscision as the outward and visible sign of his covenant with them and their pledge to be His people.
God's promise received historical immediate fulfilment in his physical descendents.. which as a christian I believe to be the Jews. ( I know that Muslims will disagree at that point). I claim the renaming of Isaac as Israel to show that.. both Genesis 26:4 and Jacob in Genesis 32:12 confirm this as does the verses in Exodus 1:7.
But.. there is another fulfilment which has an even more eternal fulfilment which is talked of in Romans chapter 9. The true descendents of Abraham are not Jews, Muslims or any other race.. but according to Romans 9
Not all who are descended from Israel are Israel. Nor because they are all descended from Abraham are they all Abraham's children. On the contrary. .... In other words, it is not the natural children that are God's children, but it is the children of the promise, who are regarded as Abraham's offspring."
What if He did this to make known the riches of His glory, even us, whom he also called, not only from the Jews, but also from the Gentiles. As He says in Hosea
"I will call them my people who are not my people.
And I will call her my loved one, who is not my loved one. Hosea 1:10
and
It will happen that in the very place where it was said to them "you are not my people" they will be called the sons of the living God".
Please read on into chapter 10 and see that the blessing given to Abraham, was that people from every tribe, tongue nation and people group would be around the throne worshipping God. (Revelation 7)
This is what is important when we think of Abraham.. are we numbered amongst his true children?