What is meant by "arbitrary" in the context of digital filter design?
The term "arbitrary" in this context refers to the desired frequency response, and what it means is that the respective design method accepts any desired response. So unlike required by many standard design routines, the magnitude response does not need to be piecewise constant or linear, and, more importantly,  the phase response need not be linear or minimum-phase. The corresponding design problem is a complex approximation problem because the desired frequency response which must be approximated is a complex-valued function.