Is it permissible for one who is financially well-off to take his sister and her son for Umrah at his own expense, knowing that her husband is unable to bear the expenses of their Umrah?

If your sister's son is an adult and of sound mind, it is permissible for you to take him with you as a mahram and take his mother for 'Umrah, and you will be rewarded for that. However, if he is young and you both have not performed 'Umrah before, it is permissible for you both to travel with him for the obligatory 'Umrah, provided there is safe company. But if you have already performed 'Umrah, it is not permissible to travel with him due to his young age, as he is not a legally recognized mahram. It is a condition for a mahram to be an adult and of sound mind to protect the woman.