Should he do Hajj on behalf of his mother or his father?

The mother’s right to being honoured is greater than the
father’s right. 

Al-Bukhaari (5971) and Muslim (2548) narrated that Abu
Hurayrah (may Allaah be pleased with him) said: A man came to the Messenger
of Allaah (peace and blessings of Allaah be upon him) and said: “O
Messenger of Allaah, who among the people is most deserving of my good
company?” 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.” 

Ibn Battaal said: What this means is that the mother is
entitled to three times more of honouring than the father. That is because
of the difficulties of pregnancy, then childbirth, then breastfeeding. This
is something that applies only to the mother and she goes through suffering
because of it. Then the father participates in raising the child. This is
also referred to in the verse in which Allaah says (interpretation of the
meaning): 

“And We have enjoined on man (to be dutiful and good) to his
parents. His mother bore him in weakness and hardship upon weakness and
hardship, and his weaning is in two years”

[Luqmaan 31:14] 

The command to be dutiful and good applies equally to both
here, but the mother is singled out with regard to these three things. 

Al-Qurtubi said: what is meant is that the mother is entitled
to a greater share of her child’s honour, and that takes precedence over the
father’s right in the case of conflict. 

‘Iyaad said: The majority are of the view that the mother
takes precedence over the father with regard to honour and good treatment.
Or it was said that they are equal in this regard. But the former view is
the one that is correct. 

End quote from Fath al-Baari. 

Al-Nawawi said in Sharh Muslim: 

The reason why the mother is given precedence here is because
she puts a great deal of effort into caring for him, shows compassion
towards him and serves him. She suffers when she carries him (in her womb),
and when she gives birth to him, and when she breastfeeds him, then she
brings him up, serves him, nurses him and so on. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was
asked a similar question and he replied: 

Do Hajj on behalf of your mother first, because the mother
has more right to being honoured than the father. This has to do with the
obligatory Hajj. But if the mother’s Hajj is naafil and the father’s Hajj is
obligatory, then start with the obligatory Hajj on behalf of the father. But
borrow money in order to send someone to perform Hajj on behalf of your
father. If next year comes and you are able to do Hajj, then do Hajj on
behalf of your father. Doing Hajj on his behalf yourself is better than
delegating someone else to do it, because your sincerity towards your father
is greater than anyone else’s. Hence we say: it is not permissible for you
to borrow money in order to delegate someone to perform Hajj on behalf of
your father. Rather you should do Hajj on behalf of your mother this year,
so long as you are able to, and next year if you are able to, do Hajj on
behalf of your father. End quote. 

Fataawa Ibn ‘Uthaymeen, 21/134.