A minor cannot perform proxy হজ্জ 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 ওয়াজিব হজ্জ. A child's হজ্জ, while valid and rewarded, is considered voluntary (nafl) and does not count as their ওয়াজিব হজ্জ. Since the child has not fulfilled their own obligation, they cannot fulfill it for someone else.
Sheikh Ibn Baz stated that the proxy for হজ্জ must be a mature Muslim (baligh) who has completed their own ওয়াজিব হজ্জ. He cited the hadith of Shubrumah, where নবী করীম told a man to perform হজ্জ for himself first before performing it for Shubrumah. If an adult cannot perform proxy হজ্জ 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 হজ্জ. A child may perform হজ্জ for themselves (as a voluntary act with reward), but they cannot serve as a representative for someone else's ওয়াজিব worship. Parents looking to send a proxy for a deceased relative should choose an adult who meets all the conditions of proxy হজ্জ.