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?

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.