If a speaker clearly emphasizes a word or a term, should it be written down in quotation marks?

Solution 1:

Italics are used for emphasis and to identify words as words.

However, in this case, big data is not really being emphasized in terms of pronunciation, nor, as the sentence is written, are the words being used as words (as they are in this sentence).

Although this is mostly a matter of style, it's currently held by most that quotation marks should only be used for actual quotes.

If sarcasm is specifically intended, scare quotes can be used—but they have come to be used in cases where sarcasm is not the actual intent, such as with "big data" (their use here was deliberate), and, so, such sentences are often rephrased.

In the case of what we call air quotes (in person, people might make a quoting gesture with their fingers as they say something—even though it's not necessarily meant sarcastically), it's better to leave the phrase alone and precede it with so-called. This avoids any ambiguity of interpretation.

The Chicago Manual of Style (17th ed.), 7.59, says:

A word or phrase preceded by so-called need not be enclosed in quotation marks. The expression itself indicates irony or doubt. If, however, it is necessary to call attention to only one part of a phrase, quotation marks may be helpful.

      So-called child protection sometimes fails to protect.
      Her so-called mentor induced her to embezzle from the company.

      but

      These days, so-called “running” shoes are more likely to be seen on the feet of walkers.

In the final example, the use of so-called almost acts like a dialogue tag for "running," turning it into a pseudo-quotation. And the quotation marks, as stated, are only used in this specific case of calling attention to the one word. (Which is subtly different from emphasizing it with italics.)

So, your example sentence can be rephrased in a couple of way:

Everyone's so intimidated by so-called big data.
Everyone's so intimidated by the term big data.