Where did the name "Argument" come from to name the angle when complex numbers are written in polar form?To the best of our knowledge, the word argument was originally used by astronomers when referring to certain angles associated with orbits. The argument of pericenter, for example, is defined as follows. There is a line through space which is fixed by convention as a reference. If one body orbits another (for instance the Moon orbiting around the Earth) the argument of pericenter associated with this orbit is the angle in the plane of the orbit between the reference line and the closest point of the orbit to the focus. The angle described above is measured from the focus of the orbit and in the direction which the object orbits.
Measuring the argument of a complex number is similar. The reference axis can be thought of as the x-axis. The location of the complex number is analogous to the closest point of the orbiting object to the focus or origin.
I do not know why "argument" was first used to mean angle or arc in astronomy. The earliest citation given in the Oxford English Dictionary is from Chaucer, circa 1391: "To knowe the mene mote and the argumentis of any planete" (Astrol. xliv. 54).