The idea in brief
A recent (2015) work, The First Nativity (Part II): History and Theology of Our Incarnate Lord and Savior by Joseph David Rhodes contains a good discussion of Neri. In short, the hypothesis is that Neri was the biological father of Shealtiel while Jeconiah was his legal father.
The idea is not entirely new (the following image is from The New Testament of Our Lord and Saviour Jesus Christ in the Original Greek published in 1861), but is especially well argued by Rhodes.
Evidence from I Chronicles
Rhodes suggests that a careful reading of I Chronicles 3:15-20 supports an interpretation that the passage is outlining legal heirs wherever it says "his son", not necessarily biological children. The repetition of the name Zedekiah in v15:
The sons of Josiah: Johanan the firstborn, the second Jehoiakim, the third Zedekiah, the fourth Shallum (ESV)
The descendants of Jehoiakim: Jeconiah his son, Zedekiah his son
Means that Zedekiah was a biological son of Josiah and brother of Jehoiakim. (See also II Chronicles 36:10). He became the legal heir of Jeconiah when
the king of Babylon made Mattaniah, Jehoiachin's [=Jeconiah] uncle, king in his place, and changed his name to Zedekiah. (II Kings 24:17)
and thus is called "his son" here. The right of succession then passed to Shealtiel, who was not a biological son of Jeconiah. Thus making Shealtiel the "son" of Jeconiah.
A plausible explanation
While we don't know the exact reason for a non-biological heir here, it is an entirely plausible scenario. For example, the law concerning Levirate marriage states:
If brothers dwell together, and one of them dies and has no son, the wife of the dead man shall not be married outside the family to a stranger. Her husband's brother shall go in to her and take her as his wife and perform the duty of a husband's brother to her. 6 And the first son whom she bears shall succeed to the name of his dead brother, that his name may not be blotted out of Israel. (Deuteronomy 25:5, ESV, emphasis added)
That is not to say that what happened here was a Levirate marraige per se. The point is that, in Jewish thought, someone could be considered the child of X without being the biological child of X.
Rhodes' hypothesis would also explain how the curse/prophesy of Jeremiah 22:28-30:
Is this man Coniah [=Jeconiah] a despised, broken pot, a vessel no one cares for? Why are he and his children hurled and cast into a land that they do not know? O land, land, land, hear the word of the LORD! Thus says the LORD: “Write this man down as childless, a man who shall not succeed in his days, for none of his offspring shall succeed in sitting on the throne of David and ruling again in Judah.”
fits in to Jesus' genealogy. If Shealtiel is only the legal successor of Jeconiah and not actual his son, there is no problem with Jesus being the Messiah.
This interpretation also fits with the theory that Luke's genealogy traces the biological lineage of Jesus (either through Mary or the biological parent in Levirate marriages), while Matthew traces His legal lineage.
While we, of course, have no way of verifying whether Neri is the biological father of Shealtiel, it does seem likely that this is what Luke is saying.