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, sallallaahu ‘alayhi wa sallam, is His slave and messenger.
If one takes a vow to do one of the acts that bring one closer to Allaah Almighty, then he should fulfill it according to the consensus of the people of knowledge. Allaah, the Exalted, praised His slaves who fulfill vows, as He said (what means): {They [are those who] fulfill [their] vows and fear a Day whose evil will be widespread.} [Quran 76:7] and (what means): {Then let them end their untidiness and fulfill their vows.} [Quran 22: 29] Moreover, the Prophet, sallallaahu ‘alayhi wa sallam, said: "If one vows to obey Allaah Almighty, then he should obey Him, and if one vows to disobey Allaah Almighty, then he should not disobey Him.” [Al-Bukhari]
In case your mother vowed to take your brother to perform Hajj and did not determine a specific timing for that by words or intention, and if she was not able to fulfill the vow this year, then she remains committed to it. Whenever she becomes financially capable and whenever your brother’s work conditions allow him to perform Hajj, she should fulfill her vow. If she specified a certain timing for doing this, be it this year or in this specific Hajj of hers, then she should fulfill it. If she is financially unable or your brother was not able to go, then she should offer an expiation like that for breaking an oath because the Prophet, sallallaahu ‘alayhi wa sallam, said: “If anyone takes a vow that he is unable to fulfill, its expiation is the same as that for breaking an oath, and if one takes a vow that he is able to fulfill, then he should fulfill it.” [Abu Daawood and Ibn Maajah]
Expiation for breaking an oath consists of either feeding ten needy people from the average of that which you feed your own family, or clothing them, or freeing a slave. Whoever cannot find any of the above should fast for three days.
Allaah Knows best.