The key to understanding is the difference between objects and names of objects:
A meronom is a part.
A meronym is the name of a part.
A meronomy is a relationship between parts and sub-parts.
Meronymy is a relationship between words.
Mereology is the mathematical study of parts and wholes. It cares about mathematical objects, not about words. The kind of mathematical structure mereology studies is a relation called a meronomy.
Why not 'mereonomy'? You do see that sometimes. But 'meronomy' is more common, and more correct, albeit also more obnoxious. You see, someone was being clever: 'Meronomy' is a back-formation from 'taxonomy'. Taxonomy is based on subsumption, rather than composition. Taxonomy is in turn the basis for set theory. Set theory and mereology are alternative ways to formulate a universal foundation for mathematics. If you're a programmer, you can think of them as the difference between relational data structures and object-oriented data structures. (If you're not a programmer, you're on your own.)
Finally, a metonym is a figure of speech that substitutes one word or phrase for another. This is superficially unrelated to meronymy and meronomy. But if a metonym happens to involve substituting a meronym for a holonym, then you have a kind of metonymy called synecdoche. Actually, though, I think the Wikipedia entries on these other terms include "Not to be confused with metonymy" based on the similar spellings, rather than this somewhat tortured connection. ;-)