Why harmonic oscillator / motion / potential is called "harmonic"? As in the topic. Why force being proportional to displacement concludes the name as being "harmonic"? Question is purely about the name.
 A: This is a good question. Harmonic vibrations (tones) actually mean sinusoidal vibrations with an integer multiple of a sinusoidal basic frequency. I did not find any explanation of the term, only that it is synonymous with an oscillator having this linear potential dependence, which of course has only one sinusoidal vibration. Perhaps one should look into an etymological dictionary. 
A: Because simple harmonic motion does this:

It just repeats, undisturbed (in harmony). It probably goes back to some Latin word to describe a musical tone(s) that repeats, indifferent to everything around it.  
Then the idea got transfered to any repeating/periodic physical process.
Dictionary definitions:
is the combination of simultaneously sounded musical notes to produce a pleasing effect.
"the piece owes its air of tranquillity largely to the harmony"
"the quality of forming a pleasing and consistent whole."
synonyms:balance, symmetry, congruity, consonance, coordination, blending, correspondence, compatibility
"the simplicity of the individual parts focused attention on the harmony of the whole structure"
Ideally, to me it just means it keeps moving evenly, for a mass on a spring say. 
But if it has modes of vibration, they are also known sometimes as harmonics, as in musical instruments like guitar strings, which have a first, second, third,.....harmonic as you restrict the movement of the string at fixed intervals
There are other kinds of harmonic motion, such  as damped harmonic motion.
