When problems happen with church officials (as they always will -- those folks are sinners like the rest of us), someone will invariably suggest that instead of electing officials, we should cast lots (draw straws, flip coins, whatever).
They give the example of Matthias, who was selected by lot to take the place of Judas, who had forfeited his apostleship by betraying our Lord.
And the Apostles were right to do this. However, just a couple of chapters later, when the first deacons were selected, we find the church "choosing" those who would take that office. Presumably by vote or consensus, but choosing, nonetheless.
What was the difference? The first event occurred in Acts 1.21-26:
"Wherefore of these men which have companied with us all the time that the Lord Jesus went in and out among us, Beginning from the baptism of John, unto that same day that he was taken up from us, must one be ordained to be a witness with us of his resurrection. And they appointed two, Joseph called Barsabas, who was surnamed Justus, and Matthias. And they prayed, and said, Thou, Lord, which knowest the hearts of all men, shew whether of these two thou hast chosen, That he may take part of this ministry and apostleship, from which Judas by transgression fell, that he might go to his own place. And they gave forth their lots; and the lot fell upon Matthias; and he was numbered with the eleven apostles."
Note the words: "shew whether of these two thou hast chosen." In other words, the purpose of the lots was to determine a choice which God had already made. Even in the midst of the lots process, criteria (a witness of the resurrection who had been with the Apostles from the time of John's baptism) were used.
The event in Acts 6 -- which I'd argue is the more relevant for our discussions about churchly officials -- is different. Men were needed for a task in the church. They were to be the first deacons, and I'd also argue that they were thereby trained to be pastors. The Jerusalem church was given direction by the Apostles: "look ye out among you seven men of honest report, full of the Holy Ghost and wisdom," and choose from such. We're not even told that the Apostles had a say in the process: the men chosen were simply set before them.
Likewise, in I Timothy 3, St. Paul counsels Timothy about the qualifications of pastors and deacons (3.1-13). Timothy is to take such criteria, and use them to choose out those who will lead the congregations.
To be honest, the whole idea of lots is usually well-intentioned, but not a good idea. God has given us abilities, intelligence, and has charged us with doing certain things. We are not Manicheans, who believe that we're somehow rocks that God just throws. We are, to use St. Paul's words, "workers together with [Christ]" (II Cor. 6.1) and we have a responsibility to use our abilities well. And that sometimes includes choosing. We should not try to be holier than God. God has commanded us, not to flip coins, but to make rational, conscious decisions, informed by God's word and the church.
I'm not making a law here. If you're in a 3 man committee, and everyone's equally qualified, there's no problem with drawing straws to see who will serve as chairman. But serving in leadership positions in God's church is important, and we should, whenever possible, use those most qualified. We might wish that God would magically inform us whom we should pick, but that's not the way God operates. Use your abilities and choose. It's God's plan.