I removed the (big soup) claim at the top of the article and revised the Etymology section.  Minestra is Italian for soup and -one is an augmentative suffix.  It could be translated as "big soup," but that is perhaps too literal.  In any case, saying that it means "big soup" does not aid understanding, but rather confuses.  Why is it big?  Is it made in large quantities?  Are the vegetables big?  And so on.  It is sufficient to say that minestra means soup and -one is an augmentative suffix.
Under Etymology, I made changes to improve conciseness.  It is not necessary to cite every etymologically related word (i.e., "minister" and "minus"); in fact, doing so clearly caused confusion since an editor in the History section of this article refers the reader to the Etymology section for the 14th c. appearance of minestrone.  Quite apart from being OR (that's another copy-edit) it is incorrect; the 14th c. is when the verb "to minister" appeared in English.  It has nothing whatever to do with vegetable soup except for deriving ultimately from the same Latin root.