Praise be to Allah.
The Sunnah urges Muslims to offer ‘aqeeqah for the newborn in general terms, without qualification.
It was narrated from Samurah ibn Jundub that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Every child is in pledge for his ‘aqeeqah which should be sacrificed for him on the seventh day, and his head should be shaved and he should be given a name.” Abu Dawood (2838) and at-Tirmidhi (1522); classed as saheeh by al-Albaani in Irwa’ al-Ghaleel (4/385).
It was narrated from Umm Kurz that she asked the Messenger of Allah (blessings and peace of Allah be upon him) about the ‘aqeeqah, and he said: “For a boy, two sheep and for a girl, one sheep.” Narrated by at-Tirmidhi (1516); he said: It is a saheeh hadith.
An illegitimate child is included in the general meaning of these hadiths, so ‘aqeeqah should be offered for him.
As he is named after his mother, his mother should offer ‘aqeeqah for him.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked:
Is it permissible for the mother to offer ‘aqeeqah for her illegitimate child, and does the child have the right to maintenance? He replied: Yes, she can offer the ‘aqeeqah and it is encouraged for her to offer ‘aqeeqah for her child, and she is obliged to spend on his maintenance if she is able to do so. If she is not able to do that, the child should be handed over to orphanages sponsoered by the state. But if she is able to do that, she should raise him, take care of him and offer ‘aqeeqah for him. She is obliged to bring him up, and to repent to Allah from what she has done, and the child is to be named after her.
The one who committed zina must repent, but he is not obliged to spend on the child’s maintenance, and the child is not regarded as being his, as he is illegitimate. He must repent to Allah, but the child belongs to the woman and is to be named after her, and she must spend on his maintenance.
End quote from Majmoo‘ Fataawa ash-Shaykh ‘Abd al-‘Azeez ibn Baaz (28/124).
For more information, please see the answer to question no. 230367 .
And Allah knows best.