Yes, a child's Hajj is valid and rewarded. Ibn ʿAbbās reported that a woman lifted up a boy to the Prophet ﷺ and asked, "Is there Hajj for this one?" He replied, "Yes, and you will have a reward" (Sahih Muslim 1336). However, the scholars are agreed that this counts as a voluntary (nāfilah) Hajj: once the child reaches puberty, the Hajj of Islam remains obligatory upon him when he becomes able, since legal obligations bind only the one who has reached maturity — the Prophet ﷺ said that the pen is lifted from three, among them "a boy until he reaches puberty" (Sunan Abi Dawud 4403), and Allah made Hajj binding on "whoever is able to find a way to it" (Quran 3:97). In practice, a young child is entered into iḥrām by his guardian, who makes the intention on his behalf, while a discerning child intends for himself with his guardian's permission; the guardian performs the rites with him and does on his behalf whatever the child cannot do himself, such as throwing the pebbles.
Q&A · Hajj
Is the Hajj of a child valid, and does it count as his obligatory Hajj?
Evidence — the daleel
On deen2u
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.