Origins
Minister means servant. The word comes from Latin minister, meaning an attendant, assistant, or subordinate, built from minus (less) with the agent suffix -ter. A minister was literally the lesser person in a relationship — the one who serves, not the one who commands. This makes it the exact structural opposite of magister (master), which is built from magis (more) with the same suffix. Servant and master, minister and magistrate: same construction, opposite meanings.
Christianity adopted the word early. Church Latin used minister for those who serve God and the congregation, preserving the humility of the original meaning. This sense entered English through Old French in the 13th century, and minister remains a standard title for Protestant clergy. The Catholic Church generally prefers priest or father, but the concept of ministry as service is shared across denominations.
The political meaning developed in parallel. Royal ministers were originally servants of the crown — officials who carried out the monarch's wishes. The title preserved the fiction of subservience even as ministers accumulated real power. By the 17th and 18th centuries, the Prime Minister was the most powerful person in the British government, despite bearing a title that literally means first servant.