In this article Hebrew scholar Michael Heiser makes the claim that the term μονογενής (monogenēs) in John 3:16 means "unique" rather than "only-begotten." However, he does not provide citations, referring to "Scholars of Greek."
Short Answer: The word is best translated "one-of-a-kind" or simply "unique". ("Only" would also work, though it could be misunderstood more easily.) The old translation "only-begotten" was based on an honest mistake in parsing the Greek word.
Background on "only-begotten"
The Greek word in question is μονογενη. It is pretty clearly a compound word formed from combining "μονο" with "γενη". Since μονο means "one", or "only", and γενη looks like the verb "γενναω", which means "to beget", the translation "only-begotten" seemed to make sense, and was the preferred interpretation for a long time.
Background on "one-of-a-kind"
However, as our understanding of the mechanics of Koine Greek have improved, it was eventually recognized that γενη cannot be a derivation of γενναω, and is instead from the word γενος (genos, meaning "kind"). If John had intended to say "only-begotten", he would have said "μονογεννητος" (single + begotten), not "μονογενης" (single + kind). (cf. VGNT).
Γενος has a semantic range including "people", "race", "descendant", or in its most basic sense, simply "kind". The term obviously isn't claiming that He is a "people", "race", or "descendant", and so the most basic sense of the word ("kind") makes the most sense here.
Thus we have the corrected interpretation of "one-of-a-kind", as supported by such instances as Psalm 21:21 (LXX) and Psalm 24:16 (LXX) where it clearly cannot be understood as "only-begotten", even if the grammatical difficulties with that interpretation were overcome.
Deliver my soul from the sword,
My only life from the power of the dog. -Psalm 21:21 (LXX) / 22:20 (NASB)
Turn to me and be gracious to me,
For I am lonely and afflicted. -Psalm 24:16 (LXX) / 25:16 (NASB)
Heiser's statement (from the document you link) is:
The Greek word μονογενής (monogenēs) doesn’t actually mean “only begotten.” It presents a problem neither with respect to Jesus having a beginning, nor with respect to divine “sons of God” who are called gods (אֱלֹהִים , elohim) in the Old Testament. The confusion extends from a misunderstanding of the root of the Greek word. For many years, μονογενής (monogenēs) was thought to have derived from two Greek terms, μόνος (monos, “only”) and γεννάω (gennaō, “to beget, bear”). Scholars of Greek eventually discovered, though, that the second part of the word μονογενής (monogenēs) does not come from the Greek verb γεννάω (gennaō), but rather the noun γένος (genos, “class, kind”). The term literally means “one of a kind” or “unique” with no connotation to time, origin or solitary existence. The validity of this understanding is shown by the New Testament itself. In Hebrews 11:17, Isaac is called Abraham’s μονογενής (monogenēs)—but Isaac was not the only son Abraham fathered, since he fathered Ishmael prior to Isaac. The term must mean that Isaac was Abraham’s unique son—the son of the covenant promises and the line through which the messiah would come. Just as Yahweh is an אֱלֹהִים (elohim), and no other אֱלֹהִים (elohim) are Yahweh, so Jesus is the unique son, and no other sons of God are like Him.
To summarize his argument:
(a) Certain Greek scholars discovered that the word μονογενής was derived the noun γένος, and not γεννάω
(b) γένος means "class" or "kind"
(c) Isaac is called Abraham’s μονογενής in Hebrews 11:17, but Isaac was not Abraham's only son.
(d) μονογενής means one of a kind or class (i.e. "unique") and nothing else
First, I would say that the argument is invalid, since the conclusion could still be false even if all the premises are true. Bauer's A Greek-English Lexicon of the New Testament and Other Early Christian Literature, for example, lists several different meanings of μονογενής in both secular and sacred Greek sources (i.e. Greek New Testament plus Septuagint) where on some occasions the word means "unique", as Heiser suggests (e.g. Cornutus - a Roman who sometimes wrote in Greek), but on many others clearly means "only child" (i.e. "only begotten"). The word may, in fact, have the etymology that certain Greek scholars "discovered", but that does not prevent the word from having other multiple meanings that may not be entirely consistent with its supposed roots.
Heiser's point about Isaac is valid and I think what he claims is correct: that the term here meant that Isaac was unique with respect to being the son through whom the Messiah would come, and not that Isaac was an only child of Abraham. The Greek texts here read:
... τὸν μονογενῆ ... πρὸς ὃν ἐλαλήθη ὅτι ἐν Ἰσαὰκ κληθήσεταί σοι σπέρμα
Heiser has a point in that many English translations render τὸν μονογενῆ here as only begotten or only son or only begotten son, which many readers would take at face value to mean "only child". But that is not what the Greek says: τὸν μονογενῆ is qualified by the phrase πρὸς ὃν ἐλαλήθη ... ("of whom was said"). The NIV goes so far as to say his one and only son, which is clearly false. A couple of versions avoid the phrase only begotten or only son. The Eastern Orthodox Bible: New Testament, for example, translates v.17-18:
By faith, Abraham, when he was tested, offered up Isaac [in sacrifice]. Yes, he who had joyfully received the promises was now offering up his uniquely loved son, the very one of whom it was said, “In Isaac will your seed be called."
The late Eastern Orthodox Bishop Dmitri Royster explains Hebrews 11:17-18:
Isaac here is called the "only-begotten" son of the one "that had received the promises." Actually, he had another son, Ishmael, born of Sara's maid Hagar (Genesis 16:15), but it was to be the son of Sara, Abraham's wife, that would be counted as the only-begotten, or "beloved son" (ton huion sou ton agapēton Genesis 22:2 LXX). God Himself confirmed Sara's insistence that Ishmael would not be the one through whom Abraham's seed would be called (Genesis 21:10-12).*
* The Epistle to the Hebrews: A Commentary (St. Vladimir's Seminary Press, 2003), p.187