Where did the idea of being "called to preach" come from?

The most direct verse I have found on this subject is Hebrews 5:4

And no man taketh this honour unto himself, but he that is called of God, as was Aaron.

In Exodes 28 it describes how Aaron was called. He was chosen out of a crown by Moses the prophet. He did not call himself but was chosen by God.

A short answer but I feel it is pretty direct. If anybody can dispute it let me know.

Where did the idea of being “called to preach” come from?

The gifts imparted to those in the early church would constitute an assumption of "calling" or divine assignment. The direction of the Holy Spirit was clear and unambiguous.

Acts 11:12 And the Spirit bade me go with them, nothing doubting. Moreover these six brethren accompanied me, and we entered into the man's house:

Acts 16:7 After they were come to Mysia, they assayed to go into Bithynia: but the Spirit suffered them not.

It seems that this direct and irrefutable intervention of the Holy Spirit became less pronounced fairly early in church history. As this happened, those seeking the specific will of God found themselves attempting to discern this in various ways.

When considering what should be done and who should do it, criteria such as skill in public speaking, scholarship in doctrinal issues, or feelings of purpose came to be used to determine God's will.

I see the use of the terms "overseer" and "elder" as being interchangeable.

1 Peter 5:1-3 The elders which are among you I exhort, who am also an elder, and a witness of the sufferings of Christ, and also a partaker of the glory that shall be revealed: Feed the flock of God which is among you, taking the oversight thereof, not by constraint, but willingly; not for filthy lucre, but of a ready mind; Neither as being lords over God's heritage, but being ensamples to the flock.

One could make a case that the oversight of a group of Christians is supposed to lay in the hands of those old enough and mature enough and less as a specific career path.

Many Christians are indeed given a special calling from the Lord Jesus Christ.

In Eph. 4:1, the apostle Paul wrote,

παρακαλῶ οὖν ὑμᾶς ἐγὼ ὁ δέσμιος ἐν κυρίῳ ἀξίως περιπατῆσαι τῆς κλήσεως ἡς ἐκλήθητε

which is translated as,

Therefore, I, the prisoner in the Lord, urge you to walk worthily of the calling (κλήσεως) in which you are called (ἐκλήθητε).

(κλῆσις = (n.) calling; καλέω = (v.) to call)

The apostle Paul will then enumerate some of the "callings" that Christians may be called to walk in.

In Eph. 4:8, it is written,

διὸ λέγει ἀναβὰς εἰς ὕψος ᾐχμαλώτευσεν αἰχμαλωσίαν καὶ ἔδωκεν δόματα τοῖς ἀνθρώποις (Textus Receptus)

which is translated as,

Wherefore, he says, "When he ascended to the high place, he led captives into captivity, and he gave gifts to men."

What were these gifts that he, the Lord Jesus Christ, gave to men (τοῖς ἀνθρώποις)?

In Eph. 4:10, it is written,

καὶ αὐτὸς ἔδωκεν τοὺς μὲν ἀποστόλους τοὺς δὲ προφήτας τοὺς δὲ εὐαγγελιστάς τοὺς δὲ ποιμένας καὶ διδασκάλους

which is translated as,

And, on the one hand, He gave the apostles, but on the other hand, the prophets, the evangelists, the shepherds, and the teachers.

The problem with Eph. 4:10 is that it contains an ellipsis, τοῖς ἀνθρώποις, which must be supplied from Eph. 4:8. In Eph. 4:8, the Greek phrase «...ἔδωκεν δόματα τοῖς ἀνθρώποις» ("he gave gifts to men") consists of the verb ἔδωκεν (same verb in Eph. 4:10), the accusative (direct object), plural δόματα ("gifts"), and the dative (indirect object), plural τοῖς ἀνθρώποις ("to men"). However, in Eph. 4:10, we encounter ἔδωκεν followed by a series of accusative plurals: τοὺς...ἀποστόλους τοὺς...προφήτας τοὺς...εὐαγγελιστάς τοὺς...ποιμένας καὶ διδασκάλους. The apostle Paul doesn't include the indirect object τοῖς ἀνθρώποις --- the individuals to whom the gifts were given --- but it most certainly should be understood and written as an ellipsis.

Therefore, it should be understood as,

And, on the one hand, He gave the apostles, but on the other hand, the prophets, the evangelists, the shepherds, and the teachers to men.

Accordingly, the δόματα ("gifts") given to men are:

• the apostles (τοὺς ἀποστόλους)
• the prophets (τοὺς προφήτας)
• the evaneglists (τοὺς εὐαγγελιστάς)
• the pastors/shepherds (τοὺς ποιμένας)
• the teachers ([τοὺς] διδασκάλους)

Those are the gifts that our Lord Jesus Christ gave to men when he ascended to heaven. In this particular verse, the apostle Paul is not saying that the Lord Jesus Christ gave gifts to the apostles, prophets, evangelists, pastors/shepherds, and teachers. That is not his point here, although it may be true elsewhere in other epistles.

John Eadie wrote,

The object of the apostle, in harmony with the quotation which he has introduced, is not simply to affirm the fact that there are various offices in the church, or that they are of divine institution; but also to show that they exist in the form of donations, and are among the peculiar and distinctive gifts which the exalted Lord has bequeathed. The writer wishes his readers to contemplate them more as gifts than as functions.

The apostle Paul then explains why our Lord Jesus Christ gave these gifts (apostles, evaneglists, etc.) to men, i.e. to the saints of the Church.

In Eph. 4:11, it is written,

πρὸς τὸν καταρτισμὸν τῶν ἁγίων εἰς ἔργον διακονίας εἰς οἰκοδομὴν τοῦ σώματος τοῦ Χριστοῦ

which is translated as,

for the perfection of the saints, for the word of the ministry, for the edification of the body of Christ.

In summary, when our Lord Jesus Christ ascended to heaven, he gave gifts to men. The gifts that were given were (1) the apostles, (2) the prophets, (3) the evangelists, (4) the pastors/shepherds, and (5) the teachers. The men to whom the gifts (enumerated above) were given were the saints of the body of Christ, i.e. the Church. The reason these gifts were given to the Church was for its perfection, for the work of the ministry, and for its edification.

There can be no doubt, then, that the offices of apostles, prophets, evangelists, pastors/shepherds, and teachers were callings or vocations. They are the very gifts that our Lord Jesus Christ bestowed upon His own Church.

In Latin the word vocare meaning "to call" gave us the English Word "vocation." A vocation is synonymous with a career (Remember vocational guidance?) in many people's minds, and it betrays the roots why someone feels "called to ministry" Both Protestants and Catholics would use these terms interchangeably.

Bibically, there is much precedence for this. Because of the terminology, I'm going to use KJV liberally:

1 Corinthians 7:20 20 Let every man abide in the same calling wherein he was called.

2 Timothy 1 8 Be not thou therefore ashamed of the testimony of our Lord, nor of me his prisoner: but be thou partaker of the afflictions of the gospel according to the power of God; 9 Who hath saved us, and called us with an holy calling, not according to our works, but according to his own purpose and grace, which was given us in Christ Jesus before the world began,

Likewise, Paul often introduces himself as "called"

Romans 1:1 Paul, a servant of Jesus Christ, called to be an apostle, separated unto the gospel of God,

1 Paul called to be an apostle of Jesus Christ through the will of God, and Sosthenes our brother,

That said, you are entirely correct in saying that such a "calling" is something to which one can "earnestly aspire."

It is because "calling" has had both senses that it came to be seen as a cooperative effort between God and Man - a gift that one would want to manifest. In practice, many churches use the terminology of "discerning a call to ministry" because of the confusion.