I'm going to have to disagree with @Barrie (a first for me on this board) and say this is an example of antithesis. The requirements for the device are somewhat looser than he would have you believe.
Let's look at a basic definition of the term from NOAD:
antithesis |anˈtiTHəsis|
noun ( pl. antitheses |-ˌsēz| )
a person or thing that is the direct opposite of someone or something else: love is the antithesis of selfishness.
• a contrast or opposition between two things: the antithesis between occult and rational mentalities.
• a figure of speech in which an opposition or contrast of ideas is expressed by parallelism of words that are the opposites of, or strongly contrasted with, each other, such as “hatred stirs up strife, but love covers all sins”: his sermons were full of startling antitheses.
Note the example "hatred stirs up strife, but love covers all sins": love and hatred are directly antithetical, true, but the actions described in each clause are certainly not diametric opposites, and it is the clauses, not their subjects, that form the elements of the antithesis.
Now, turning back to the Shelley quote:
How much happier that man is who believes his native town to be the world, than he who aspires to be greater than his nature will allow
This is a somewhat veiled antithesis, to be sure: there are no obvious contrast pairs like love and hate that distinguish the clauses. Nevertheless, it is the meaning of each clause — what it points to — that furnishes the material for contrast. In the first case we have a man who "believes his native town to be the world" — in other words, who is content with what he has at hand, and does not therefore overreach; in the second we have a man who does overreach by aspiring to be "greater than his nature will allow." The antithesis here arrives in the contrast between wise contentment and rash ambition: not overreaching vs. overreaching.
If we put this fragment into the context of Frankenstein, it informs the entire import of the novel. Dr. Frankenstein was the exemplar of overreaching, as he sought (in Shelley's view) to become as God and create life. It was the kind of prideful act that is regularly cast down in Greek tragedy, and the resultss of Frankenstein's act, you will recall, were disastrous.
It's a sentence, not a phrase, but I don't think that is the answer your teacher is looking for.
"Guts" -- which literally means "intestines" but is used to mean courage -- is an idiom, as you suspected, or more exactly a figure of speech.
"Pure guts" is hyperbole, an exaggeration for rhetorical effect.
"Steel" is a metaphor ("the use of a word or phrase to refer to something that it is not, invoking a direct similarity between the word or phrase used and the thing described").
Perhaps the technique referred to is one of these.
Best Answer
I'm assuming the previous sentence in the discourse explained some other change to her (e.g., her hair changed color). The prepositional phrase with the influence of her dress functions an adjunct.
In English adjuncts often consist of a prepositional phrase, but not always. So you should cast your net broader than just prepositional phrases. Consult the relevant Wikipedia entry for more details.