The original meaning of「商」can be roughly deduced from its structure.「商」was originally comprised of「辛」(a chisel) on top of「丙」(a place where some kind of foundation to do a job is set up, e.g a table):
The total combination had something to do with a workshop-type job, such as clothes tailoring:
「口」was added later, as a distinguishing mark for the dynasty name of Shang. Note that a similar practice occurred with the character「周」for the Zhou Dynasty.
「商量」, one of the primary meanings of「商」, had something to do with measuring (e.g. measuring materials required for the job), extended to mean discuss, approximate, calculate. The latter meanings can be wrapped up in with a generic meaning like extracting a result from thinking or talking over a problem, which is where the meaning of quotient originated from. This meaning resulted in one of the specialist jargon used in Rod calculus being labelled as「商」. For details, see the Wikipedia section on using rod calculus to approximate the square root of 234567, where the row labelled "Shang" is「商」. Its modern meaning as quotient in division originated from this term.