Should he do Hajj on behalf of his deceased father who did Hajj, or on behalf of his elderly mother who has not yet done Hajj?
Question

                I performed the obligatory Hajj last year, praise be to Allah. 

My mother has not done Hajj and she is over 65 years of age. Because of her health and the situation in Egypt, she is not able to do Hajj. 
My father died this year at the age of 80; he did Hajj twenty years ago. 

Is it better for me to do Hajj on behalf of my father, for whom I hope for speedy forgiveness from Allah, and I do not want to wait until next year, or should I do Hajj on behalf of my mother who has not done Hajj before? 

Is it essential for me to take money from her wealth for the expenses of Hajj? Or is it permissible for me to pay for it, even though my mother has enough money for that? 

I asked my mother to be pleased with me in this world and in the Hereafter, and never to be angry, and I asked my brothers to bear witness to that, knowing that I always try to make her pleased with me.
Praise be to Allah.
      If your mother is unable to do Hajj herself for a reason 
      for which there is no hope of an end, and she has enough money to do Hajj, 
      then she should appoint someone to do Hajj on her behalf. If you volunteer 
      to do Hajj on her behalf, that is part of honouring her, and it is not 
      essential that the expenses of Hajj be paid from her wealth. 

    See the answer to question no.
    36841. 

    With regard to whether your Hajj this year should be on 
    behalf of your father or your mother: 

    The answer is that you should do it on behalf of your mother. 
    That is for two reasons: 

    1.Hajj on behalf of your mother 
    is obligatory, because she has not done Hajj before, whereas Hajj on behalf 
    of your father would be naafil (supererogatory), and what is obligatory 
    takes precedence over what is naafil.

    2.If there is a conflict between 
    the rights of the father and the rights of the mother, then the rights of 
    the mother take precedence, because she has three times the rights of the 
    father. That is indicated by the report narrated by al-Bukhaari (5971) and 
    Muslim (2548) from Abu Hurayrah (may Allah be pleased with him) who said: A 
    man came to the Messenger of Allaah (blessings and peace of Allah be upon 
    him) and said: Which of the people is most deserving of my good 
    companionship? He said: “Your mother.” He said: Then who? He said: “Then 
    your mother.” He said: Then who? He said: “Then your mother.” He said: Then 
    who? He said: “Then your father.” 

    Al-Qurtubi said: What is meant is that the mother is entitled 
    to the greater share of the son’s kindness, and she takes precedence in the 
    case of a conflict with the father’s rights. 

    ‘Iyaad said: The majority are of the view that the mother 
    takes precedence over the father with regard to the son’s kindness. And it 
    was said that their rights are the same, but the former view is more 
    correct. 

    End quote from Fath al-Baari, 10/402 

    With regard to your saying “I asked my mother to be pleased 
    with me in this world and in the Hereafter, and never to be angry, and I 
    asked my brothers to bear witness to that, knowing that I always try to make 
    her pleased with me”, there is nothing wrong with your asking her to be 
    pleased with you in this world. But with regard to asking her to be pleased 
    with you in the Hereafter, this is something that cannot be guaranteed in 
    this world; it is something that one cannot be certain about. 

    The Hereafter is something unknown (ghayb) and no one 
    can be certain about anything with regard to it. 

    So continue striving to please her and honour her until you 
    attain that. 

    With regard to your father, your du‘aa’ for him is better 
    than doing Hajj on his behalf, so offer a lot of du‘aa’ for him. 
    “and say: ‘My Lord! Bestow on them Your Mercy as they did 
    bring me up when I was young’” [al-Isra’ 17:24]. 

    And Allah knows best.