Religious sense of "minister"
The word minister has the general meaning of "one who acts upon the authority of another" (see here).
According to the Online Etymology Dictionary (see this entry), it is from c. 1300 and derives from the Latin word with the same spelling, which means "servant, priest's assistant".
The meaning related to religious functions is therefore prior to the Protestantism, which began in the 15th century only.
The Wikipedia has a page that explains the meaning of the word in the Catholic Church as:
In the Catholic Church the term minister enjoys a variety of usages. It most commonly refers to the person, whether lay or ordained, who is commissioned to perform some act on behalf of the Church. It is not a particular office or rank of clergy, as is the case in some other churches, but minister may be used as a collective term for vocational or professional pastoral leaders including clergy (bishops, deacons, priests) and non-clergy (theologians and lay ecclesial ministers). It is also used in reference to the canonical and liturgical administration of sacraments, as part of some offices, and with reference to the exercise of the lay apostolate.
The Wikipedia also brings definitions of the word in different domains (see this page):
Minister (Christianity), a Christian who ministers in some way
Minister (diplomacy), the rank of diplomat directly below ambassador
Minister (government), a politician who heads a ministry (government
department)
At the very core of its meaning, a minister is simply someone who serves others (see @kiamlaluno's answer), whether it be under the government (public/civil servants), the Lordship of Christ (servants of God/the Most High/etc), or any other authority or mandate. Today, the word minister may unfortunately conjure up an image of superiority or high office, but the fact remains that ministers are [supposed to be] servants, usually with special tasks to fulfill or accomplish.
The NOAD reports the following definition of the noun minister.
- (also minister of religion) a member of the clergy, especially in Protestant churches.
- (also minister general) the superior of some religious orders.
- (in certain countries) a head of a government department: Britain's defense minister.
- a diplomatic agent, usually ranking below an ambassador, representing a state or sovereign in a foreign country.
- archaic a person or thing used to achieve or convey something: the Angels are ministers of the Divine Will.
As per the origin of the word, the dictionary reports that the word, when used for the sense 1 and 3, derives from the Old French ministre (noun), ministrer (verb), which derives from the Latin minister "servant," from minus "less."