Can the mother do the ‘aqeeqah for her child if the father has divorced her?

Firstly: 

The ‘aqeeqah is Sunnah and mustahabb, but it is not obligatory. The one who follows this Sunnah will attain the reward and virtue, and the one who does not follow it will be falling short but he does not incur sin. This is the view of the majority of scholars, as has been explained previously in the answers to questions no. 162021, 20018 and 38197. 

Secondly: 

The basic principle is that it is prescribed for the ‘aqeeqah to be done from the wealth of the father of the child, not from the wealth of the mother or the wealth of the child himself, because it is the father who is primarily addressed in the hadeeths which say that ‘aqeeqah is prescribed. 

But the fuqaha’ said: It is permissible for someone other than the father to offer the ‘aqeeqah on behalf of the child in the following cases: 

1.If the father falls short and refuses to slaughter the ‘aqeeqah

2.If permission has been sought from the father to offer the ‘aqeeqah on his behalf and the father has given permission.

They quoted as evidence for that the proven report from Ibn ‘Abbaas who said: The Messenger of Allah (blessings and peace of Allah be upon him) offered ‘aqeeqah on behalf of al-Hasan and al-Husayn (may Allah be pleased with them), two rams for each. Narrated by al-Nasaa’i (no. 4219) and classed as saheeh by al-Albaani in Saheeh al-Nasaa’i. 

They said: The fact that the Prophet (blessings and peace of Allah be upon him) offered the ‘aqeeqah on behalf of his grandsons al-Hasan and al-Husayn (may Allah be pleased with them) indicates that it is permissible for the ‘aqeeqah to be done by someone other than the father if it is done with his permission and his consent. 

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said, commenting on the hadeeth, “Every child is in pledge for his ‘aqeeqah, so (the sheep) is to be slaughtered for him on the seventh day, and his headis to be shaved, and he is to be given his name” (narrated by Abu Dawood, 3838; classed as saheeh by al-Albaani in Saheeh Abi Dawood):  

The fact that the verb slaughter appears in the passive form that the one who offers the ‘aqeeqah is not specified. According to the Shafaa‘is, it is the duty of the one who is obliged to spend on the child. According to the Hanbalis, it is the duty of the father unless he is unable to do that because he is deceased or he refuses to do it. 

Al-Raafa‘i said: In this hadeeth it is as if there may have been a reason why Prophet (blessings and peace of Allah be upon him) offered ‘aqeeqah for al-Hasan and al-Husayn. 

Al-Nawawi said: It may be understood as meaning that at that time his parents were poor or he volunteered to do it with the father’s permission. Or it may be that he issued instructions that this be done, or it may be one of the things that are unique to the Prophet (blessings and peace of Allah be upon him), just as he offered the udhiyah on behalf of those of his ummah who had not offered the sacrifice. Some of them regarded it as one of the things that were unique to him. End quote from Fath al-Baari, 9/596 

To sum up: 

It is not obligatory for the mother to offer the ‘aqeeqah on behalf of the newborn; rather it is mustahabb for her to do that if the father refuses to do it or if the father is unable to offer the sacrifice because he is far away or he is unaware of the birth and so on, and Allah will record the reward for her. 

Please see also the answer to question no. 71161 

Thirdly: 

With regard to reciting the adhaan in the ear of the newborn, there is no saheeh hadeeth concerning that. Some of the fuqaha’ said that it is mustahabb. This has been discussed previously in the answer to question no. 136088 

Imam Maalik (may Allah have mercy on him) stated that this action is not mustahabb. 

If we say that it is prescribed to recite the adhaan in the ear of the newborn, as is the view of the Shaafa’is and others, the more correct of the two opinions – in sha Allah – is that it is permissible for a woman, whether it is his mother or another Muslim woman, to do that. This is different from the opinion of those scholars who stipulated that this should be done by a man, as is the case with regard to the adhaan for prayer. 

Al-Shabramalsi al-Shaafa‘i (may Allah have mercy on him) said: 

The words “it is Sunnah to recite the adhaan” mean even if it is done by a woman, because this is not the adhaan that is to be done by men; rather the aim here is simply to mention the name of Allah and seek blessing.

End quote from his commentary (haashiyah) on Nihaayat al-Muhtaaj, 8/149. 

This is what it says in Haashyat al-Shuburi ‘ala al-Manhaj, that it is not essential that the person who recites the adhaan in the ear of the newborn should be male. This is supported by the view of some shaykhs, that this Sunnah is fulfilled by the midwife reciting the adhaan in the ear of the newborn.

End quote from Haashiyat al-Tablaawi ‘ala Tuhfat al-Muhtaaj, 1/461 

And Allah knows best.