How is a New Church President Called?
It's interesting to realize that the Church and its members don't call a prophet, they call a president. It is by the laying on of hands and the bestowal of keys that the President of the Church becomes the Lord's Prophet.
And again, the duty of the President of the office of the High Priesthood is to preside over the whole church, and to be like unto Moses — Behold, here is wisdom; yea, to be a seer, a revelator, a translator, and a prophet, having all the gifts of God which he bestows upon the head of the church. (D&C 107:91-92)
Understanding how a new president is called requires understanding how authority works in the general councils of the Church. It is a testimony of the Restoration that the Lord has provided for this process.
If the President is dead, who has authority to lead the Church?
The twelve traveling councilors are called to be the Twelve Apostles, or special witnesses of the name of Christ in all the world—thus differing from other officers in the church in the duties of their calling. And they form a quorum, equal in authority and power to the three presidents previously mentioned. (D&C 107:23-24)
The Seventy are also called to preach the gospel, and to be especial witnesses unto the Gentiles and in all the world—thus differing from other officers in the church in the duties of their calling. And they form a quorum, equal in authority to that of the Twelve special witnesses or Apostles just named. (D&C 107:25-26)
The First Presidency automatically dissolves with the death of the president, just as every presidency in the Church is automatically dissolved with the death and/or release of its president.
This means that in the absence of the President and the First Presidency, the Quorum of the Twelve Apostles is automatically the presiding authority of the Church until a new President can be called and a new First Presidency organized.
In the case of an extreme emergency (such as the improbable death of all the Apostles), the Quorum of the Seventy are equal in authority with the Twelve and can restore both the Twelve and the First Presidency of the Church.
But, to exercise this authority, the quorums must act unanimously. (Conditions were provided should the quorum in its entirety could not meet, which makes sense in the 1800s with quorum members traveling internationally.)
And every decision made by either of these quorums must be by the unanimous voice of the same; that is, every member in each quorum must be agreed to its decisions, in order to make their decisions of the same power or validity one with the other — A majority may form a quorum when circumstances render it impossible to be otherwise — Unless this is the case, their decisions are not entitled to the same blessings which the decisions of a quorum of three presidents were anciently, who were ordained after the order of Melchizedek, and were righteous and holy men. (D&C 107:27-29)
And with that authority, the Twelve can call a new President
While these verses explain that the quorums have authority when acting unanimously, they don't specifically grant authority to reconstitute the First Presidency. The Lord provided for this.
It is the duty of the Twelve, also, to ordain and set in order all the other officers of the church, agreeable to the revelation which says: To the church of Christ in the land of Zion, in addition to the church laws respecting church business — verily, I say unto you, saith the Lord of Hosts, there must needs be presiding elders to preside over those who are of the office of an elder; and also priests to preside over those who are of the office of a priest; and also teachers to preside over those who are of the office of a teacher, in like manner, and also the deacons — wherefore, from deacon to teacher, and from teacher to priest, and from priest to elder, severally as they are appointed, according to the covenants and commandments of the church. (D&C 107:58-63)
Then comes the High Priesthood, which is the greatest of all. Wherefore, it must needs be that one be appointed of the High Priesthood to preside over the priesthood, and he shall be called President of the High Priesthood of the Church; or, in other words, the Presiding High Priest over the High Priesthood of the Church. From the same comes the administering of ordinances and blessings upon the church, by the laying on of the hands. (D&C 107:64-67)
The Quorum of the Twelve Apostles has the authority to call and set apart a new president to the Church and the First Presidency. They do this by laying on hands and giving the new President the keys of presidency, revelation, seership, translation, and prophecy, which they have being equal in authority from the preceding President according to verses 91-92, which we have already read.
But, who do they call?
Of necessity there are presidents, or presiding officers growing out of, or appointed of or from among those who are ordained to the several offices in these two priesthoods. Of the Melchizedek Priesthood, three Presiding High Priests, chosen by the body, appointed and ordained to that office, and upheld by the confidence, faith, and prayer of the church, form a quorum of the Presidency of the Church. (D&C 107:21-22)
The Twelve have the authority to call anyone who is ordained to the Melchizedek Priesthood. However, since the call to Brigham Young in 1847, the tradition of the Quorum has been to extend the call to the senior member of the Quorum of the Twelve Apostles.
But if any member of the Melchizedek Priesthood could be called, why do we call the senior Apostle?
It actually took a few presidents to clearly establish the tradition of calling the senior member of the Quorum of the Twelve. We would need to read some citations from the Journal of Discourses to cover that, but it actually isn’t unreasonable that the Quorum settled on this tradition.
Organize yourselves; prepare every needful thing; and establish a house, even a house of prayer, a house of fasting, a house of faith, a house of learning, a house of glory, a house of order, a house of God; yhat your incomings may be in the name of the Lord; that your outgoings may be in the name of the Lord; that all your salutations may be in the name of the Lord, with uplifted hands unto the Most High. (D&C 88:119-120)
The Church has the authority to organize itself to meet its needs and thus the Quorum of the Twelve have the authority to determine the rules of succession so long as they meet the commandments of scripture. As the D&C teaches, we need not be commanded in all things (D&C 58:26-28). It's better that we make choices for ourselves, and the Twelve decided to automatically extend the call of President of the High Priesthood to the senior member of the Quorum of the Twelve Apostles, providing for a predictable and orderly transition of authority within the Church when a President passes away.