Al-Jawab Billahi At-Taufeeq (the answer with Allah’s guidance)
To be Baligh, (to be physically matured) is one of the conditions that make Hajj compulsory. In the case of the male, it is from when the signs of puberty appear, e.g. wet dream, growing of beard, etc. If there are no signs, then from the age of 15 lunar years.
In the case of a female, it begins at menses or if there is no menses then at 15 lunar years.
Regarding Hajj for a child:
“Muslim scholars agree that Hajj is not obligatory upon those who have not reached the age of puberty, because the Pen is lifted from children, as the Prophet (Sallallahu Alaihi Wasallam) said: “The Pen is lifted from three (i.e., their deeds are not recorded): a child until he reaches puberty; an insane man until he comes to his senses; and one who is asleep until he wakes up.” (Abu Dawud, 4403; Ibn Majah, 2041)
If a child performs Hajj, then his/her Hajj is valid and he/she will be rewarded for it, InshaAllah. However, this Hajj will not count as the obligatory Hajj.
Ibn `Abbas, (RA) reported that the Prophet (Sallallahu Alaihi Wasallam) met some riders in al-Rawha’ and said, “Who are these people?” They said, “Muslims.” They said, “Who are you?” He said, “The Messenger of Allah.” A woman lifted up a child and said, “Is there Hajj for him?” He said, Yes, and you shall have a reward.” (Muslim, 1336.)
Therefore, they will be obliged to do the Hajj e Fardh again once they reached maturity.
Also, if a woman wanted to perform Hajj and she has sufficient wealth and a husband, or any Baligh (mature) Mahram relative must accompany with her. if she does not have any *mature Mahram relative to accompany her then it is not permissible for her to go for Hajj.
Even if, till her death no Mahram is available, she cannot go for Hajj, rather it becomes Fardh upon her to make Wasiyyat of a Hajj-e-Badal. (Ahsanul Fataawa part 4 pg.532)
*Note: A Mahram as young as 15 years of age can generally qualify as a mature Mahram.
And Only Allah Ta’ala Knows Best.
Moulana Qamruz Zaman