Praise be to Allah.
If your mother does not have the money to do Hajj, then Hajj is not obligatory in her case, because Allah, may He be glorified and exalted, says (interpretation of the meaning):
“And Hajj (pilgrimage to Makkah) to the House (Ka'bah) is a duty that mankind owes to Allâh, those who can afford the expenses (for one's conveyance, provision and residence)”
[Aal ‘Imraan 3:97].
If the son wants to send his mother for Hajj, the scholars differed as to whether it is permissible to pay for that from zakaah. The reason for this difference of opinion is that Allah listed, among the categories on which zakaah may be spent, “and for Allâh's Cause” [al-Tawbah 9:60]. Does this include Hajj or does it apply only to jihad for Allah's cause?
Most of the scholars are of the view that Hajj is not included in the categories on which zakaah may be spent.
Ibn Muflih al-Hanbali (may Allah have mercy on him) said:
It (zakaah) cannot be given for Hajj. According to one report, he favoured this view in al-Mughni and classed it as sound in al-Sharh. This was also stated by most of the scholars, because the phrase “Allah’s cause (sabeel-Allah) usually refers to jihad, and zakaah is only given to those who need it, such as the poor, or those whom the Muslims need, such as the worker who collects and distributes zakaah. The Muslims do not benefit from one individual’s Hajj and have no need for it; it is not obligatory in the case of the poor, for whom this duty is waived, and if a poor person still wants to do it, it is better to spend this amount of money on those who are in need or to spend it on helping the Muslims.
Al-Mubdi‘ Sharh al-Muqni‘, 2/387
Imam Ahmad has another view, which is that it is permissible to give zakaah to someone who wants to do Hajj. This view was also favoured by Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him), and is mentioned in fatwas issued by the scholars of the Standing Committee for Issuing Fatwas.
This has been discussed previously in the answer to question no. 40023.
Based on this second opinion, it is permissible for you to send your mother for Hajj using the zakaah of your wealth. But if you take the more cautious approach and do not regard this as zakaah, that is preferable, and we hope that Allah will accept it from you and from her.
And Allah knows best.