It is said that 'Ali, may Allaah be pleased with him, was born in the Ka'bah and that he opened his eyes in the arms of the Prophet, sallallaahu alayhi wa sallam. How true is this claim?

All perfect praise be to Allaah, The Lord of the Worlds.  I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

We are not aware of any evidence in the Book of Allaah or the Sunnah (Prophetic tradition) of the Prophet, sallallaahu ‘alayhi wa sallam, that 'Ali was born in the Ka'bah. However, it is confirmed that Hakeem ibn Hizaam is the Companion who was born in the centre of Ka'bah. [Muslim] 

Besides, there are no scholars who mentioned that 'Ali opened his eyes in the arms of the Prophet, sallallaahu ‘alayhi wa sallam.

Nonetheless, 'Ali has many merits and deeds of excellence, and there is no need to look for merits that are not authentically confirmed about him.

Allaah Knows best.