What is the intention if the family of the deceased wishes to donate or give charity on his behalf? Should the intention be "for our deceased father" or "for the sake of Allah Almighty for our deceased father"? And what intention has been established in the Prophetic Sunnah?

The intention of someone giving charity on behalf of the deceased is to dedicate the reward of the good deed to a specific person, as Ibn Qudamah stated in Al-Mughni: "And any act of worship he performs, and dedicates its reward to a Muslim deceased person, will benefit him—Allah willing." The wording of the dedication is broad, and what matters is the intention, for indeed, actions are judged by their intentions.