Is it permissible for a mother to breastfeed her cousin's daughter so that she becomes a sister to her children, even though there is no need for it?

Your mother may breastfeed her sister's daughter if the parents permit it, thereby making her a mahram to her children. However, it is not permissible to adopt her and attribute her lineage to your family, as adoption is forbidden in Islamic law, based on the Almighty's saying: "Call them by [the names of] their fathers; it is more just in the sight of Allah. But if you do not know their fathers, then they are your brothers in religion and your allies. And there is no blame upon you for that in which you have erred, but [there is blame for] that which your hearts intended. And ever is Allah Forgiving and Merciful."