What is the best way to engage in ongoing charity (Sadaqah Jariyah): buying a small house for orphans or contributing to the construction of a mosque, when selling a residence and donating half of it or half its value for the sake of Allah?

Perpetual charity (ṣadaqah jāriyah) is an endowment (waqf), which means tying up the principal and dedicating its usufruct. Building a mosque falls under endowment. As for buying a house for orphans, if it is for the purpose of transferring ownership to specific orphans, then it is a charity and not an endowment. However, if it is to allow orphans to benefit from living in it without transferring ownership, such that the residence passes to another orphan when their orphanhood ceases, or if it is to rent it out and spend its proceeds on orphans, then it falls under endowment. The endowment should be for poor orphans, not wealthy ones, as jurists have differed on the validity of endowments for the wealthy, and Sheikh al-Islam Ibn Taymiyyah favored its invalidity. The preference between building a mosque and providing housing for poor orphans depends on the need in each place.