...(among others), covers this topic and related aspects of Islam's past, in both "A History of God" and her book on Mohamed. Most of what she writes about it refers to pre-Islamic legends that have been sewn into the tapestry of this religion's belief system over time. Here's another perspective on the origins of The Rock:
According to ancient Arabian traditions, when Adam and Eve were cast from Paradise they fell to different parts of the earth; Adam on a mountain on the island of Serendip, or Sri Lanka, and Eve in Arabia, on the border of the Red Sea near the port of Jeddah. For two hundred years Adam and Eve wandered separate and lonely about the earth. Finally, in consideration of their penitence and wretchedness, God permitted them to come together again on Mt. Arafat, near the present city of Mecca (previously called Becca or Bakkah, meaning narrow valley).
Adam then prayed to God that a shrine might be granted to him similar to that at which he had worshipped in Paradise. Adam's prayers were answered and a shrine was built. (This is a pre-Islamic legend and the Koran, the Islamic Holy Scripture, says nothing whatsoever of Adam’s connection with Mecca or of a shrine he prayed at). Adam is said to have died and been buried in Mecca and Eve in Jeddah by the sea which still bears her name, jiddah, meaning maternal ancestor in Arabic.
This shrine passed away during the era of the flood,
at which time the body of Adam began to float on the water while the Ark of Noah circumambulated around it and the Ka’ba seven times before journeying north where it landed after the flood. A thousand years later, according to one Islamic tradition in 1892 BC, the great patriarch of monothesism, Abraham, or Ibrahim, came to Mecca with his Egyptian wife Hagar and their child Ishmael. Here Hagar lived with her son in a small house, at the site of the earlier shrine, and Abraham came to visit her on occasion.
http://www.sacredsites.com/middle_east/saudi_arabia/mecca.html In Armstrong's version, she repeats the legend that the Ka'ba was in fact the place where Adam supposedly "fell." But she writes further that it was later used as the place where pre-Islamic Arabs built their shrines with statuettes of the various gods from their pantheon that they worshiped. That would include the gods of Babylonia (who dominated almost all the Semetic nomadic peoples of that era) and who would include the Babylonian goddess Ishtar (whom the Jewish Kabbalists referred to as Lilith - Adam's first wife who refused to be submissive -- which is my favorite goddess as you know) and other gods from the Akkadian/Sumerian pantheon.
All this god and goddess worshiping would include the polytheistic worship by Mohamed himself, as did almost all Arabs at the time. But one of those "minor gods" that was worshiped and that Ms. Armstrong mentions in her book, is a minor god whose name was Allah. She writes that they also circumambulated around the rock calling out prayers and thanks, etc., to their favorite gods back then -- just as they still do to the only one they've got left, today.
- K&R