Praise be to Allah.
The historians and some of the mufassireen have stated that her name was Rahmah bint Meesha ibn Yoosuf ibn Ya’qoob.
But this is something which is not proven in any clear sound text, rather it was transmitted from the books of the People of the Book, or by some Muslims from them. We shall list those who were of this view and transmitted it:
1 – Al-Suyooti said: Ibn ‘Asaakir narrated that Wahb ibn Munabbih (may Allaah be pleased with him) said: “The wife of Ayyoob (peace be upon him) was Rahmah (may Allaah be pleased with her) bint Meeshaa ibn Yoosuf ibn Ya’qoob ibn Ishaaq ibn Ibraaheem (peace be upon them).
(al-Durr al-Manthoor, 7/197. Also in Tafseer al-Baydaawi, 3/310; Tafseer al-Qurtubi, 9/265; Tafseer al-Baghawi, 2/451)
2 – Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:
But these Israa’eeli (i.e., Jewish) reports to be mentioned for the purpose of proving a point, they are not to be believed. They are of three types:
(i)Those which we know are sound because of evidence which we have which proves them to be true. These are saheeh.
(ii)Those which we know are false because of evidence which we have which contradicts them.
(iii)Those which we do not know whether they are true or false. So we do not believe in them and we do not disbelieve in them, but it is permissible to narrate them for the reasons given above.
Most of them are things which serve no religious purpose, hence the scholars of the People of the Book differ greatly concerning such things. The mufassireen also differed concerning them as a result of that, as they mentioned, for example, the names of the People of the Cave, the colour of their dog and their number, or what kind of tree the staff of Moosa came from, and other matters which Allaah did not mention in detail in the Qur’aan because knowing the specific details does not serve any worldly or religious purpose.
(Majmoo’ al-Fataawa, 13/366-367).
Al-Shanqeeti (may Allaah have mercy on him) said:
What the mufassireen have said concerning the name of their dog – some said that its name was Qitmeer, and some said that its name was Hamdaan, etc. – we need not dwell on at length, because it serves no purpose. There are many things in the Qur’aan which neither Allaah nor His Messenger has explained to us in detail, and there are no proven reports concerning them; there is no benefit to be gained by researching such matters.
(Adwaa’ al-Bayaan, 4/48)
And Allaah knows best.