Short Answer: It's Significance Leads to some Ambiguity in Focus
There are two pertinent syntactical factors here1
First, Daniel Wallace notes, nouns as objects of prepositions (ἐν here) are a case where regularly definiteness is inherent even though the article is lacking. Here is his statement, but then note what is in the midst of this about Heb 1:2...
There is no need for the article to be used to make the object of a
preposition definite. However, this is not to say that all
prepositional objects are definite. An anarthrous noun as object of a
preposition is not necessarily definite. It is often qualitative
(e.g., ὑιῷ in Heb 1:2, mentioned above), or even occasionally
indefinite (cf. μετὰ γυναικὸς ἐλάλει—"he was speaking with a woman"
[John 4:27]). Thus, when a noun is the object of a preposition, it
does not require the article to be definite: if it has the article, it
must be definite; if it lacks the article, it may be definite. The
reason for the article, then, is usually for other purposes (such as
anaphora or as a function marker).
So second, Wallace believes the anarthrous use here in Heb 1:2 is to express qualitative purposes. The note he refers to as "mentioned above" regarding Heb 1:2 is this:
Although this should probably be translated "a Son" (there is no
decent way to express this compactly in English), the force is clearly
qualitative (though, of course, on the continuum it would be closer to
the indefinite than the definite category). The point is that God, in
his final revelation, has spoken to us in one who has the
characteristics of a son. His credentials are vastly different from
the credentials of the prophets (or from the angels, as the following
His note about "on the continuum" is in reference to an image in his book showing the three ways anarthrous nouns can function, with overlap of qualitative at each end of the spectrum to the other two:
Here is Wallace's statement about what qualitative is:
A qualitative noun places the stress on quality, nature, or essence.
It does not merely indicate membership in a class of which there are
other members (such as an indefinite noun), nor does it stress
individual identity (such as a definite noun). It is akin to a
generic noun in that it focuses on the kind. Further, like a generic,
it emphasizes class traits. Yet, unlike generic nouns, a qualitative
noun often has in view one individual rather than the class as a
Wallace seems to not be following his own ideas here. Note the last statement again, "unlike generic nouns, a qualitative noun often has in view one individual rather than the class as a whole," and yet of the Heb 1:2 passage, he stated in the second quote above that "it would be closer to the indefinite than the definite category" (which is then emphasizing class, not individual!).
Further Analysis Leading More to Definiteness
The problem is that as a "class," the individual Christ is one of a kind. That means all the qualitative aspects merge and are expressed only in one individual, and thus point to someone very definite. After all, there is no other "son" δι᾽ οὗ καὶ τοὺς αἰῶνας ἐποίησεν ("through whom also He made the worlds," NKJV), as v.2 itself notes.
So at the very least, Wallace (in my opinion) should have hedged his qualitative argument toward the definite side of the continuum. But I myself would lean toward agreement with the many translations that insert "His" before "son" (KJV, ESV, NASB, NIV, NKJV, etc.), which are correctly making it into a very definite reference (i.e. that the qualitative is not in view as much as the specific individual). The whole context of Heb 1:1-2:8 is leading up to the naming of this son in 2:9, "Jesus," which is a very definite reference. And again, the qualities noted in those verses are only true of this One "son," so the definite identity is not distinguishable into a "class" of individual different from this "specific" individual.
So the conclusion then is that being anarthrous is not really very significant, because clearly there is only one definite individual being noted here.
1 Quotations taken from the discussion in Daniel B. Wallace, Greek Grammar Beyond the Basics - Exegetical Syntax of the New Testament (Zondervan Publishing House and Galaxie Software, 1999), 243-247.