Tarshish is Tarsus of Cilicia.

It has long been popular to argue for more exotic or distant identifications for Tarshish, including by myself in the past.  But Josephus said Tarshish was Tarsus, and this modern article backs up that identification convincingly.

Tarshish (Josephus' Tarsus Reconsidered)
http://www.bibleorigins.net/tarshishtarsus.html

There is one detail in that I can't agree with, and that's arguing for a late dating of Genesis based on pre Esarhaddon Assyrian Inscriptions rendering Tarsus was Tarzi.  It could be Tarzi was a mistaken Assyrian form corrected by later Assyrians who knew more directly what they called themselves, or the Tarzi inscriptions could be scribal errors.  There is no need to question the reliability of Genesis over this.

Identifying Tarshish with Tarsus also best fits the thesis of this post of mine from last year, (though in said post I also considered a Cretan identification).
https://mithrandironchronology.blogspot.com/2018/12/what-does-greek-even-mean.html

Those are all very technical and scholarly reasons for that identification.  What I want to speculate on now is how it could theologically serve the Meta-narrative of The Bible to connect Old Testament Tarshish to New Testament Tarsus.

Like OT Tarshish it's never a location the narrative visits directly (same with Cilicia as a whole), the few times it seems like Acts is about to go there it then skips forward.

NT Tarsus is only relevant for being the hometown of Saul later known as Paul.  Before he's ever mentioned by name he's probably one of the Cilicians refereed to in Acts 6.

We know from Extra Biblical sources that Tarsus of Cilicia was a port city associated with sea trade and thus with ships, but Biblcially the New Testament never directly mentions that.  However Paul does spend a lot of time on ships, some travel by ship was a part of all four missionary journeys, most famously his ship wreck on Malta.

Tarshish was a grandson of Japheth, but the name is also duplicated as a Benjamite in 1 Chronicles 7:10.  I think it's possible the Chronicler is more using this name as a stand in for a Benjamite clan that would in the future live in Cilicia, possibly as a result of the slave trade alluded to in Joel 3.

Paul who was a Benjamite can be viewed as playing a role in how the Genesis 9 eschatological relationship between Japheth and Shem was fulfilled.

A number of Prophecies also speak of Ships of Tarshish playing a role in how exiled Israelites are brought back to the promised land.  In some views that too is arguably fulfilled partly by the work of Paul.