A minor cannot perform proxy Hajj on behalf of another person according to the consensus of scholars. This is because one of the essential conditions for a valid proxy is that the proxy must have already performed their own obligatory Hajj. A child's Hajj, while valid and rewarded, is considered voluntary (nafl) and does not count as their obligatory Hajj. Since the child has not fulfilled their own obligation, they cannot fulfill it for someone else.
Sheikh Ibn Baz stated that the proxy for Hajj must be a mature Muslim (baligh) who has completed their own obligatory Hajj. He cited the hadith of Shubrumah, where the Prophet told a man to perform Hajj for himself first before performing it for Shubrumah. If an adult cannot perform proxy Hajj until they have done their own, then a child — who has not even reached the age of obligation — certainly cannot serve as a proxy.
Sheikh Al-Fawzan confirmed that maturity (bulugh) is a condition for the proxy, as is having previously completed one's own Hajj. A child may perform Hajj for themselves (as a voluntary act with reward), but they cannot serve as a representative for someone else's obligatory worship. Parents looking to send a proxy for a deceased relative should choose an adult who meets all the conditions of proxy Hajj.