I dont think your question can be answered entirely, because it would just be speculation.

Any follower of Islam would claim it to have been built by Ibrahim, specifically his grandson Nabit. In the Islamic belief the Ka'bah was built by Ibrahim, lost to paganism and returned to Allah through Muhammad/Islam.

It is also well known (through the Bibliotheca Historica, by Diodorus Siculus) that the Ka'bah was a pagan shrine, sanctuary, monument etc. before it became a Islamic one and the Hajj to the Ka'bah was already a long time tradition in the Arabian peninsula before the rise of Islam. So it could have been built by pagan Arabians in Mecca before the rise of Islam.

"The Middle East" by William Ochsenwald & Sydney Nettleton Fisher.