Does a child's Hajj count?

Two separate things get confused here: whether a child's Hajj is *valid* and whether it *counts as the obligation*.

It is valid. Children were brought on Hajj in the Prophet’s ﷺ lifetime, the rites they perform are accepted, and the reward is written for them and for the parent who brought them.

It does not discharge the obligation, because the obligation only attaches to someone who is adult and able:

“In it are clear signs such as the standing place of Abraham. And whoever enters it i.e., the Ḥaram shall be safe. And due to Allah from the people is a pilgrimage to the House - for whoever is able to find thereto a way. But whoever disbelieves i.e., refuses - then indeed, Allah is free from need of the worlds.” (Qur’an 3:97)

A child is not yet addressed by that. So someone who went at seven and becomes able at forty performs Hajj then as their obligatory one. That is not a waste of the first trip; it was a voluntary Hajj, which is exactly what a repeat pilgrimage is for anyone.

Practically, for a parent considering it:

• The guardian makes the intention on behalf of a child too young to make it themselves, and performs on their behalf what the child cannot — the throwing of the pebbles, for instance.
• The child is put in ihrām and the ihrām restrictions apply to them in a way suited to their age.
• There is no minimum age, and no obligation to bring children at all.

Worth adding, because it is the real question behind this one: Hajj is physically hard, extremely crowded, and long. Scholars have not discouraged bringing children, but many experienced people advise against bringing very young ones for reasons of safety rather than of ruling. That is a judgement about your family rather than a matter of law.

The reward for a Hajj performed well is stated in absolute terms: whoever performs it without obscenity or wrongdoing returns as on the day his mother bore him (Sahih al-Bukhari 1521).