Learn English – apply to or for, and apply to for

Usually prepositions start a phrase that ends in a noun and starts from something else, where the preposition indicates the relationship, as in “all roads lead to Rome”, “good for the goose”, “a man of all seasons.” But in English, prepositions can be used in many ways that don’t follow that pattern, such as in phrasal verbs. Another different pattern is in your example sentence, which also occurs here:

When choosing an occupation, you should consider salary of secondary importance.

The “of” works the same in this sentence:

Salary is of secondary importance.

“Of secondary importance” is the subject-complement of “salary”. This is just the ordinary way that “to be” predicates one thing of another. It follows the same pattern as “Salary is important” and “My car is red.” “Of secondary importance” might seem unusual if you’re not accustomed to seeing an adjectival phrase starting with “of”. You can’t say:

Salary is secondary importance.

because “secondary importance” is a noun, and it would be absurd (factually, not grammatically) to say that salary is secondary importance. So you have to resort to “of” in its meaning of “within things that have this quality”. So: “Salary is within the category of things with secondary importance.”

In these examples, “consider” also works in its ordinary way, by predicating a complement of its object, like this:

I consider salary the most important thing in life.

“Salary” is the object of “consider”, and “the most important thing in life” is the object-complement. So, that means the same as:

I think that salary is the most important thing in life.

Probably what you found confusing in the original example was just that the object-complement starts with “of”.

Which ODO definition?

The ODO lists this sense of “of” under “Phrases”, sense 1. However, fluent speakers don't think of “of” as having neatly distinct senses like those in a dictionary, or that this sense of “of” is phrasal while the others aren’t. Constructions where “of” starts a predicate, like “of secondary importance”, “of poor breeding”, “of red color”, “of help”, etc., seem to fluent speakers like a reasonable way to apply “of”’s basic meaning of “coming from” in a situation analogous to many others where a preposition introduces a predicate, like “to the manner born”, “off his rocker”, “for rent”, etc.

Rather than memorizing a definition, I think it would be more useful to understand the grammatical pressures that lead someone to choose this construction. I think there are two main pressures:

1. You want an adjective but the language only provides a noun, and the noun wouldn't make sense in context. For example, suppose you want to say that Helen’s boyfriend has a certain crude manner about him, suggesting that he comes from a bad family or is innately crude. You can't say “Helen’s boyfriend is poor breeding,” because that doesn’t make sense. The phrase “poor breeding” seems to capture just what you want to say, but it’s a noun for a person’s family background, not an individual person. “Of”'s basic meaning of “coming from” fits perfectly, though, and since “of” is a preposition, it’s a wonderful choice to convert a noun into a predicate: “Helen’s boyfriend is of poor breeding.”

2. The prep+noun-as-predicate construction echoes familiar phrases. For example, “If there can be a man of all seasons, why not a woman of all seasons? Why couldn’t anyone or anything be of all seasons?” If you hear the phrase “of all seasons” enough, you start treating it as a unit, and eventually you feel comfortable using it as a predicate after “be”. I chose the phrase “of all seasons” to illustrate this, not because people use it frequently or use it in this construction, but because it’s just familiar enough that you could pry it out of its familiar context and use it as a predicate and reasonably expect that most people would get the reference. Also, the familiar and authoritative phrase is actually “a man for all seasons”, but the transition to “of” is so natural that a person barely notices. On the other hand, a familiar phrase like “for rent”, as in “This house is for rent”, is so familiar, you can’t easily vary it, but you can treat it as an indivisible grammatical unit and expect to be understood.

There are still other grammatical pressures that can lead a person to choose this construction, of course, like wanting the greater emphasis that comes from a noun rather than an adjective, and subtly distinguishing your meaning from the ordinary meaning of an adjective, as in “The walls were of a red color”, but the above two explain its main uses. More importantly, understanding the above pressures shows you how English grammar works mainly by reasonably varying and extending familiar constructions to fit unique and unpredictable circumstances. Understanding those pressures might be of help in countering the tendency to think that English grammar works by applying precise rules and definitions that stay the same in each circumstance (an easy misconception to get if you try to learn English from grammar books that are filled with rules).

You are correct in thinking of is usually used with Ministry

The Ministry of Silly Walks

but it may also depend on how the title is phrased

Ministry for the Propagation of Virtue and the Prevention of Vice

Ministry of Propagation.... would not sound correct.

I think your question is actually a more general question about the preposition rules for using either "Ministry" or "Society" or "Department" or "Office"

Certain wordings could only use for

Ministry for the Advancement of...
Ministry for the Preservation of...
Ministry for Homeland Defense...
Ministry for Transport
Ministry for Use of Solar Power

and others could go either way

Ministry of Trade
Ministry for Trade

Ministry of Silly Walks
Ministry for Silly Walks