Q:

Word that means "tendency to avoid something"

I am looking for a word that means "tendency to avoid something".  At first I thought that aversion fit the bill, but then I learned that it meant "a strong dislike towards something." 
For example, most people have an aversion towards feces.  I am looking for a word that more along the lines of "they don't hate PBJs, they just try to avoid them as much as possible." 

A:

Aversion is an acceptable word for this.   Read definition number 3.   
Aversion is a strong dislike, but it carries a sense of repellence, too.    Per Merriam-Webster:  It is a tendency to extinguish a behavior or to avoid a thing or situation and especially a usually pleasurable one because it is or has been associated with a noxious stimulus. 
I think this sounds exactly like the word you are requesting. 
You can absolutely have an aversion to PB&Js.   

A:

disinclination might fit.  It covers both unwillingness and a mild aversion so you tend to avoid that thing.  It does not suggest a strong dislike also. 

a lack of inclination; a mild aversion or reluctance. 

that toward which you are inclined to feel dislike

A:

Eschew is to habitually avoid for both moral/practical reasons
A list of synonymous choices; I prefer forbear
http://law.academic.ru/31912/eschew

