This is my current understanding, though it could be incorrect:
-Before Islam, most Arabs were nomadic pagans (though there were Jewish and Christian tribes)
-Islam is an Abrahamic religion, which reveres Abraham, Moses, Jesus and other figures from the Old and New Testaments
-Arab Muslims would consider themselves descended from Ishmael, Isaac's brother, the son of Abraham
Did pagan Arabs see themselves in relation to Abraham and Ishmael? Was an association with this line of descent restored by Muhammad, the Koran and Islam?
This is an interesting question. A quick look through some of my books didn't reveal much from pre-Islamic Arab sources about Abraham. I did, however, come across this passage taken from Genesis:
*These are the descendants of Ishmael, Abraham’s son, whom Hagar the Egyptian, Sarah’s slave-girl, bore to Abraham. These are the names of the sons of Ishmael, named in the order of their birth: Nebaioth, the firstborn of Ishmael; and Kedar, Adbeel, Mibsam, Mishma, Dumah, Massa, Hadad, Tema, Jetur, Naphish, and Kedemah. These are the sons of Ishmael and these are their names, by their villages and by their encampments, twelve princes according to their tribes. (This is the length of the life of Ishmael, one hundred thirty-seven years; he breathed his last and died, and was gathered to his people.) They settled from Havilah to Shur, which is opposite Egypt in the direction of Assyria; he settled down alongside of all his people. Gen. 25:12-18 (NRSV)
So an association of Arabian desert-dwellers with Ishmael and Abraham dates back at least this far. Some pre-Islamic Arabs surely would have been aware of this, so hopefully someone else can provide information about their self-perception.