Why don't they just say 'people' on board, why souls? What is the origin of this term? I'm thinking it comes from sailing as I think I've heard that term in reference to crews out at sea, but I'm not a sailor so I don't know.
The primary reason is probably that it ensures there is no confusion between passengers, crew, or infants. Technically, "passengers" is the number of seats occupied, "crew" is both the pilots and flight attendants on duty. So any small children brought on as "lap children" will not be included in the "passengers" count, but should be included in the total number of people on board.
I found another interesting point over on the English Stack Exchange site, which is that dead bodies are sometimes transported as well. In this case, some might consider these "people" as well. Also, in an incident, the bodies should not be confused with the regular passengers.
So, "souls" effectively communicates the number of living humans on board.
There may certainly be holdovers from the maritime influences on aviation as well.
I agree with what fooot said. Also, I would add, as someone to volunteers in search & rescue (Civil Air Patrol) as a mission pilot, when you hear the word "souls," it adds some urgency and seriousness to the handling of any emergency. When an air traffic controller asks a pilot, during an emergency, for the number of souls on board, it communicates to the pilot that the controller and pilot are focusing extra hard together on solving the emergency successfully, and that one word tells the pilot that the controller is going to be marshalling resources to help in every way possible. "Souls" is a term full of life and caring. For rescuers, it communicates very quickly the total number of persons who must be found and saved.
The earliest reference I can find to 'souls' as a count of persons is mid-eighteenth century, although it probably was in use earlier. It appears in maritime commerce, as the number of living humans aboard a ship, and in civics, as the population of a town or city.
I think that in the early 1700s the words 'people' and 'person' both had strong connotations compared to those words today. 'People' meant humans of a certain country or a specific culture, and 'persons' meant humans of note or important characters.
So I conclude that a need arose, driven by government and commerce, for a word to mean an unaffected and precise head count, that yet afforded those being counted a little more respect than the barrels, boxes, coins, and cows whose numbers were also tallied.
Now if I could find a reference to that usage of 'souls' from the thirteenth or fourteenth century, I would simply assume that it arose from the learned churchmen who were doing the counting, as literacy was not yet widespread.
The term Soul appears many times throughout scripture, Hebrew translation of this particular use and meaning is literally "breather." Some are on the right track with anatomical heart as this is what is required to circulate oxygen. This also clarifies why transporting deceased persons are not counted among SOB. They are no longer breathing.
The term "soul" once meant a whole person: body and spirit. Thanks to Hollywood's love affair with the word, most now believe it means spirit.
The sad result of this is several phrases now have a different meaning than the historical writings that contain them once intended.
"not one soul was saved" - everyone died, now means they are all dammed
"bless his soul" - may he have a long life (literally blessing the integrity of the body and spirit), now means bless his spirit
"soul mate" - a person physically and spiritually matched, now means a single special someone that is primarily a match of the spirit
"he sold his soul" - he has given his life to evil, now means a state of evil that primarily affects the after life
"bare one's soul" - to share the intimate details of once physical and spiritual life, now means to reveal primarily emotional details as a stand-in for spiritual things
"rest his soul" - implying that sleep is symbolic of death, now solely a blessing on the spirit
"lost soul" - a person not doing well physically and spiritually in this life, now means a person not following spiritual paths