What is the etymology of the term "amplitude" as used in quantum mechanics? Some people claim that the single most important conceptual point in all of quantum mechanics is that "probability amplitudes" (inner products) can be complex, as opposed to strictly positive. But in the earlier theory of classical wave mechanics, an inherent feature of wave amplitudes is that they are always positive. So from this perspective, calling a quantum-mechanical inner product a probability "amplitude" seems like uniquely misleading terminology. What is the historical origin of the term?
 A: As stated in the comments amplitude is a term used in descriptions of sinusoidal functions, where the y variable is the "amplitude" in various descriptions  and the x may be space or time. It is used in describing waves in general, as waves  are =modeled with differential equations called wave equations.


A sinusoidal curve
  
  
*
  
*Peak amplitude
  
  
  2.Peak-to-peak amplitude 

  
*Root mean square amplitude 
  
  
  4.Wave period (not an amplitude)

The usage in physics follows the convention of wave descriptions.You ask:

So from this perspective, calling a quantum-mechanical inner product a probability "amplitude" seems like uniquely misleading terminology. What is the historical origin of the term?

The historical reason comes from the interference effects seen in the probability density distributions from quantum mechanical systems, clearly seen as interference patterns in double slit experiments, characteristic of wave solutions.

Particularly the single electron or photon at a time, established that the sinusoidal solutions of the quantum mechanical equations and the rule of using the complex conjugate squared for describing the probability of measuring a particle at an (x,y,z) was a solid prediction, and that the postulates for choosing the quantum mechanical solutions of boundary conditions problems were valid. 
