Question: I would like to ask about the ruling of Hajj al-badal (Hajj on behalf of someone else). Did the prophet (PBUH) mentioned it? And should the children of a deceased perform it on behalf of their motheror father or mother, or can they ask someone else to do it?

Answer: Hajj al-badal has two types:
1. Hajj of chidren and grandchildren on behalf of their parents or grandparents. This type is agreed upon by scholars and their are evidences on it from the authentic sunna.

A woman told the prophet (PBUH)that her father was an old man and not able to go for hajj and she asked him if she could perform Hajj on his behalf and the propeht (PBUH) said: "Yes, perform hajj on his behalf" 
2. Hajj on behalf of someone other than parents... Scholars have different opinions regarding this kind of Hajj. Some scholars say it is ok while others don't accept it because acts of worship require intention. Imam Malik say that it is ok if the deceased had it as a will, in this case, he has the intention.
In both cases there is a condition that the person on whose behalf hajj is performed didn't ignore hajj while having the ability to perform it. He rather couldn't perform it for a legal execuse.