So, I understand what the idiom actually means, roughly -- "the final word," "the word from on high." But where does it come from? Is there a story or a history that gives it context?
It is part of a longer saying 雀の千声鶴の一声, which means that rather having than the 1000 voices of commoners (represented by 雀) having a single voice from on high (represented by 鶴) is preferable.
The reason for using 鶴 is apparently because they have a long neck and their cry is extremely high pitched cry, which overpowers the sounds in the surroundings.
This expression appears to be the second part of the proverb:
三省堂 大辞林 or
The king's word is more than another man's oath.
Cranes, like geese and so on will all take flight at once if just one of them sound the alarm. So, 鶴の一声 references this quick obedience of the many to the voice of just the one.