Tamid, Chapter 1, Mishnah 2. Each day's service in the Beit Hamikdash opened with the terumat hadeshen: one of the kohanim would scoop up a measure of the ashes lying uppermost on the mizbeach and deposit them on the pavement of the azarah, the Temple courtyard. Our mishnah describes the way this kohen was chosen, and the preparation that qualified him for the privilege.
According to the mishnah, a kohen who wanted to be a candidate for clearing the ashes would get up long before dawn, go down and immerse, and only afterward dress in the bigdei kehunah, the priestly garments.
Why the immersion? Because no one was permitted to enter the azarah without first going to the mikveh. Even a kohen who was already tahor was required to immerse before entering. So a kohen hoping to merit the terumat hadeshen would begin his day in the mikveh, and only afterward dress himself and wait.
All of this had to be completed, in the words of the mishnah, "ad shelo yavo hamemuneh", before the memuneh arrived. He needed to be immersed and dressed well in advance of the memuneh's arrival at the Beit Hamoked, the Hall of Fire that the previous mishnah described.
Who is the Memuneh?
Memuneh, the appointed one, is another title for the segan kohen gadol, the deputy who served as second in command to the kohen gadol. He oversaw a great many matters in the Beit Hamikdash, and among them was this: he was the one who arrived to designate the kohen who would carry out the terumat hadeshen.
When did he show up? No hour was fixed for it. Sometimes he came at kriat hagever, the crowing of the rooster near daybreak, sometimes a little before that and sometimes a little after. Since nobody could know in advance, a kohen who had his heart set on this service made sure to be awake, immersed and dressed as early as he could manage.
The Knock and the Lottery
The segan would arrive and knock at the door of the Beit Hamoked. The kohanim would open it from the inside, and he would enter and announce that whoever had already immersed should come forward and join the lottery to determine who would perform the terumat hadeshen. All of this took place in the Hall of Fire.
How was such a lottery conducted? For various parts of the avodah the kohanim would hold a payis. The kohanim stood in a circle around the officiating kohen, each one extended a finger, and a number was announced. He would then begin counting from one of them around the circle until he reached that number. It was, in effect, the Beit Hamikdash's version of a counting-out game played with fingers. Whichever kohen the count landed upon was the one chosen for the service.
So the mishnah concludes: "hefisu", they cast the lot, and the kohen who won it went on to perform the terumat hadeshen.