Answer
Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- Islam gives the wife the mahr which is her exclusive right, and it permits her to dispose of her own wealth.
2- It is not obligatory for a husband to bear the costs of his wife’s Hajj, even if he is rich.
3- It is recommended for him to do so and he will be rewarded for that, but he is not sinning if he does not do that. That is not enjoined by either the Quran or the Sunnah.
Answering your question, the Fatwa Center at Islam Q and A, states:
Islam enjoins on the husband to spend on his wife in a manner that is good and reasonable, but it does not oblige him to pay off her debts, or to pay zakah on her behalf, or to pay the costs of Hajj etc.
Sheikh Ibn Uthaymeen was asked: will a husband be rewarded if he appoints someone to do Hajj on his wife’s behalf if she dies without having done Hajj?
He said: it is better for him to do Hajj on her behalf himself so that the rituals will be done properly. Then he said: But as for it being obligatory, that is not obligatory for him. (Al-Liqa’ al-Shahri, 34, no. 579)
Just as it is not obligatory to make up Hajj on her behalf after she dies, so too it is not obligatory for him to pay for her Hajj whilst she is still alive.
This is as far as the issue of whether or not it is obligatory is concerned. With regard to being kind to her, if he does that then Allah does not cause the reward of those who do good to be lost, and Allah will record for him the reward of her Hajj.
Muslim jurists stated that it is obligatory for a husband to spend on his wife’s Hajj in the event that he deliberately spoils her Hajj, such as one who forces his wife to have intercourse before the first stage of exiting ihram, etc.
Sheikh Abd al-Kareem Zaydan said:
It is not one of the wife’s rights over the husband that he should bear the costs of her Hajj, or share the costs with her. (Al-Mufassal fi Ahkam al-Mar’ah, 2/177)
Sheikh Al-Albani (may Allah have mercy on him) was asked about the same issue and he replied: It is not obligatory for a husband to pay the expenses of his wife’s Hajj. If a woman has sufficient money to go for Hajj, then it is obligatory for her to go for Hajj. If she does not have enough money, then she is not obliged to go for Hajj.
Almighty Allah knows best.