Mus’ab Ibn Abdillah narrates the lineage of Hakeem bin Hizam. He added that his mother is Faakhta bint Zuhayr bin Asad bin Abdul Azza. She gave birth to Hakeem in the Ka’ba. She was pregnant and her labour pains started whilst she was in the Ka’ba. She gave birth to him in the Ka’ba. During the birth, she was on a leather mat. Her clothes under her were washed at the well of ZamZam. No one before Hakeem bin Hizam or no one after him was born in the Ka’ba.

 As regards the issue of ‘Ali may Allaah be pleased with him being born inside the Ka’bah, then this is not correct. Ibn Al-Mulaqqin may Allaah have mercy upon him said in his book "Al-Badr Al-Muneer": (Notice: Hakeem (Ibn Hizaam) was born inside the Ka’bah and there was no one else known to be born inside it, as regards what was reported about ‘Ali may Allaah be pleased with him being born inside it, then this is a Dha’eef [weak] narration).

Moreover, As-Suyooti may Allaah have mercy upon him in his book "Tadreeb Ar-Raawi", classified the chain of narrators of the narration which states that ‘Ali may Allaah be pleased with him was born inside the Ka’bah as Dha’eef. So, he disagreed with the author of Al-Mustadrak (i.e. Al-Haakim may Allaah have mercy upon him) regarding this narration and emphasized that Hakeem Ibn Hizaam was the person who was born inside the Ka’bah.

Imam Hakim (RA) states that Mus’ab ibn Abdillah had doubt at the end of the Hadith because the narrations are narrated with Tawatur (in such numbers that to deny the narration is impossible) that Hazrat Ali RA was born in the Ka’ba. Having said that, many scholars are of the opinion that the narration regarding Hazrat Ali RA being born in the Ka’ba are weak. It is the opinion that he was born near the birthplace of The Prophet Muhamad SAW.

References: Fatawa Darul Uloom Zakariya, Page, 122 – 125, Vol 1, ZamZam Publishers.

Only Allah knows best

http://daruliftabirmingham.co.uk/home/was-hadrat-ali-ra-born-in-the-kabah/
https://www.islamweb.net/en/fatwa/171581/the-person-who-was-born-inside-the-kabah