Appoint Elders
New Church Congregation
There is an incident in the book of Titus that dealt specifically with this dilemma:
The reason I left you in Crete was that you might straighten out what was left unfinished, and appoint elders in every town, as I directed you. (Titus 1:5)
It is apparent that when establishing a new local church (pioneer church) it was customary for an experienced minister to do the "appointing" (ordaining, NIV footnote). It is only reasonable, since all the congregation are New Believers, and not yet credible or qualified to be making decisions as Church elders, making decisions on their own.
In fact, the Apostle Paul listed right after his admonition to Titus, the qualifications that an aspiring Elder should have. (1:6-16) It would take some time to examine each man to see if he fit these qualifications. As an Overseer, Titus could discern who those were who could then qualify.
A list of these qualifications was also given in the book of 1 Timothy (3:1-7).
Established Church
However, in a local church that has been around for a period of time, it was more common for the presiding Elders to work together in appointing (ordaining) a new elder. This is mentioned in the case of Timothy's entrance into the ministry:
Until I come, devote yourself to the public reading of Scripture, to preaching and to teaching. Do not neglect your gift, which was given you through a prophetic message when the Body of Elders laid their hands on you. (1 Timothy 4:13-14)
In modern terminology, this is called the laying on of hands by the Presbytery. And along with a prayer, there is usually an admonition and/or even a prophetic utterance with it! This prophetic Word gives direction as far as which type of ministry he is "gifted with."
Some elders are gifted with preaching; others with evangelism. Still others, with counseling, or social work, or teaching. Some are even anointed with the gift(s) of healing! (1 Corinthians 12-14)
Confirmation
Since the prospective elders are from the local church--and must meet the qualification of an elder--many churches require a confirmation of the Elders' choice by the local Congregation. The local people have "rubbed shoulders" with the man, and so they would know if he matches the qualifications of an Elder.
But the established Elders would be first in choosing the man because they have served as overseers, keeping watch over the people for some period of time.
To the saints in Christ Jesus...together with the Overseers, and deacons. Grace and peace to you... (Philippians 1:1)
Postscript
It is manifest in the New Testament church that there were appointed a plurality of elders. This is for several good reasons (1) planning with several inputs, (2) checks and balances, (3) a variety of available ministries, etc.
