Is dropping the verb "was" an option?

Given the following examples:

• Yesterday, you mentioned thinking it was a good idea to go sailing.
• Last week, you thought it was appropriate to dress like a pirate.

Can was be inferred and therefore omitted for the sake being succinct without loss of meaning?

• Yesterday, you mentioned thinking it a good idea to go sailing.
• Last week, you thought it appropriate to dress like a pirate.

Could someone provide the grammatical terms related to my question it would be greatly appreciated as I was uncertain of what search phrases would apply.

Yes, you can infer the verb be without confusion or problem in your sentences.

The verb be is probably the most inferred verb in language. It is (and has been) such a commonly understood and inferred verb that in Latin, the be verb (am, are, is) can be dropped from the sentence completely without loss of meaning.

Le Train de Nulle Part (The Train from Nowhere, - Michel Thaler, 2004) is a 233-page novel; notable as an example of "constrained" writing, the entire novel is written without a single verb. Here is an excerpt:

What luck! A vacant seat, almost, in that train. A provisional stop, why not? So, my new address in this nowhere train: car 12, 3rd compartment, from the front. Once again, why not?

The inferred verbs (and pronouns) are (most likely)

What luck! (there is) A vacant seat, almost, in that train. (we're making) A provisional stop, why not? So, my new address in this nowhere train (is): car 12, 3rd compartment, from the front. Once again, why not?

A review:

New book, 233 pages, quite dull, no economics. French of course.

The inferred verbs (and pronouns) are (most likely)

(There is a)New book, (it is) 233 pages, (it is)quite dull, (it contains)no economics. (it is) French of course.

So you see how easy it is to drop the inferred verb be.

If someone can provide the grammatical terms related to my question it would be greatly appreciated as I was uncertain of what search phrases would apply.

Perhaps search on "extraposition" (and maybe also predicative complements). An extraposition construction has one more position in it than the corresponding more basic alternant.

Your second set of examples seem to be using extraposition, which involves the process where a subordinate clause is moved to the end and then the subordinate clause's original location is then occupied by the dummy pronoun "it".

I'll bracket the involved complements in your two examples, and bold both the dummy pronoun "it" and the relocated (extraposed) subordinate clause:

• 1b) Yesterday you mentioned thinking [it] [a good idea] [to go sailing].

• 2b) Last week you thought [it] [appropriate] [to dress like a pirate].

Those two examples have the same meaning as the following two simpler versions, which don't have the extraposition, and which happen to be ungrammatical:

• 1b.x) (*) Yesterday you mentioned thinking [to go sailing] [a good idea].

• 2b.x) (*) Last week you thought [to dress like a pirate] [appropriate].

Notice how the bracketed complements form an (object) predicand and predicative complement pair. That is: "to go sailing" is "a good idea", and "to dress like a pirate" is "appropriate".

The predicands ("to go sailing" and "to dress like a pirate") are non-finite clauses -- in this case, they are infinitivals. But an infinitival cannot occur between a verb and a predicative complement (page 1255, CGEL); and so, normally, the infinitival is extraposed. And that is what happens and results in your two versions #1b and #2b.

Compare your two examples to examples that don't involve a subordinate clause:

• She thought Tom a fool.

• She considered the idea foolhardy.

For predicands that are noun phrases, extraposition is generally not an option:

• (*) She thought it a fool Tom.

• (*) She considered it foolhardy the idea.

Those above two examples are ungrammatical.

But in examples like those in the OP's original post, which involved infinitival clauses in the object predicand location, extraposition is obligatory.

EDITED-TO-ADD: Your first pair of examples uses a declarative content clause:

• 1a) Yesterday you mentioned thinking [(that) it was a good idea to go sailing].

• 2a) Last week you thought [(that) it was appropriate to dress like a pirate].

which also uses extraposition, where it involves the subject of a content clause. That is, the more basic alternants for the content clauses are: "to go sailing was a good idea", "to dress like a pirate was appropriate".