All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The vow that this woman made falls under the category of vows that are made to do a permissible act. Muslim scholars held different opinions regarding whether this kind of vow is considered valid and binding for the person or not. The Hanbali scholars believed that such a vow is valid and binding but that the vow-maker has the choice either to deliver on it or offer expiation like for breaking his oath; this is based on Qiyas (i.e. analogy) between this kind of vow and an oath. Ibn Qudaamah said, "Making a vow to do a lawful act such as saying, 'I make a vow to Allaah, the Exalted, that I will wear my garment or ride my mount,' is similar to oath-making. The speaker has the choice to either fulfill it or to offer expiation like for breaking an oath." Al-Mardawi said, "This is the adopted opinion of the Hanbali school of fiqh, and it is held by the majority of the Hanbali scholars." [Al-Muqni‘]
Other scholars, such as the Maaliki, Shaafi‘i, and Hanafi scholars believed that such a vow is not considered a valid, binding vow. Ibn Nujaym said, "When one makes a vow to commit a sin, it is impermissible to fulfill the vow, and when he vows to do a permissible act such as eating, drinking, or having lawful sexual intercourse, then that is not considered a valid, binding vow." [Al-Bahr Ar-Ra’iq]
Ash-Shirbeeni said, "The most correct opinion in this regard is that it is not incumbent on the person to offer expiation for the vow in this case because it is not considered a valid, binding vow. This opinion is also deemed sound by An-Nawawi in his book Al-Majmoo." [Mughni Al-Muhtaj]
The second opinion is the preponderant one according to our view, Allaah willing.
Hence, it is permissible for this woman to change her mind about naming her child after her brother, and she is not required to do anything in this regard. If she offers expiation like for breaking an oath, then that would be better so as to avoid the difference of opinion among scholars.
Allaah Knows best.