In the KJV of the Hebrew Bible any time you see Greece or Greek or Grecia or Grecian it's a reference to Javan son of Japheth, his descendants and where they settled. Which scholars of the ancient Bronze/Iron Age and early classical antiquity know refereed to the Ionians and Ionia.
However in the New Testament those words are translations of references to the Hellenes. The Hellenes as an ethnic term didn't always include everyone we today mean by Greeks. However the word was also associated with anyone who could speak the Hellenic language and is thus used of Hellenized Jews like the Seven Deacons in Acts 6.
The Ionians were Hellenes, but only one of a number of Hellenic tribes. And of the mythical founder figures of the original four Hellenic tribes, Ion's parentage is questionable, which may be a mythical memory that the Ionians were not Hellenes originally but became absorbed into them. And the same may be true later on of the tribes said to descend from Hellen's sisters' sons.
The ancient root of the words Greek and Greece is one of those nephews of Hellen, Greacus, but the Hellenes typically said to come from him were mainly those of Southern Italy and Sicily, Magna Graecia. The Latins called all Hellenes Greeks after them and thus via the Vulgate that's why there are so many Greeks in our English Bibles.
Ionia was a part of Asia Minor/Turkey, south of Mysia and west of Phrygia, it included the cities of Miletus, Ephesus and Smyrna, and Islands like Samos. The Ionians also colonized more Aegean islands and eventually came to the actual mainland of modern Greece and by the early Classical period a significant portion of the population of Athens/Attica were Ionians, maybe even the majority. But the Athenians of legends set in the Heroic Age were not likely to have been Ionians.
Athens becoming the cultural capital of Greece during the Classical Period is the main basis for treating Ionian and Hellen as synonyms, but on purely genealogical grounds it doesn't work so well. Alexander The Great can be called of Javan based on his culture, being educated by Aristotle made him in many ways more Athenian then Macedonian in his way of thinking. But also Daniel 8:21 arguably describes Alexander as a King of Javan in a way that doesn't' necessarily make him Javanite himself. Daniel 11 doesn't directly link Alexander to Javan at all but mentions Javan as where the Persian Wars started, the first conflict of which is known as the Ionian Revolt.
Now I do believe that the Grandsons of Noah had more children then just the ones named, the sons named are founders of offshoot nations. Therefore I do think Ionian strictly speaking doesn't include the nations of Javan's four sons. So does that justify expanding Javan's descendants to include all Hellens? Well that's complicated, there is plenty of reason to think all their original locations were also either in modern Turkey or on islands not far from Turkish coast-lands.
Kittim is well known to refer to Cyprus, Kition, but perhaps not even all of it as Kition was one of ten ancient kingdoms of Cyprus. Attempts to expand it to being synonymous with "Greece" start with desiring to see Isaiah 23 as about Alexander's siege of Tyre. But Kittim is not actually identified as Tyre's enemy here but a place Tyrians would try to flee to, which can potentially apply equally to both Alexander's siege and Nebuchadnezzar's. Daniel 11:30 is used to try and make Kittim into Rome because in the traditional view this is where Rome starts becoming a thorn in the side of Antiochus Epiphanes, but that encounter between Antiochus and Rome happened on Cyprus. In my alternative view that this is about the reign of Ptolemy IV this may have to do with how Cyprus was under Ptolemaic control. There was even a brief dynastic connection.
Elishah is the big factor in trying to make all Hellenes into Javanites, sometimes by arguing the name Hellen itself comes from Elishah which is pretty tortured, as well as Elysium and the Elysian fields which were underworld locations, and Josephus said the Eleans (people of Elis) came from Elishah. However the Elishah of Javan is probably the Alashiya, another kingdom of ancient Cyprus, perhaps specifically the sites of Kalavasos and Alassa. Elishah and Kittim are both mentioned together with Mizraim in Ezekiel 27:7-8, at the time Ezekiel was writing Egypt was also colonizing Cyprus, or in one theory I've considered had been exiled there.
The Dodanim is most likely supposed to be read Rodanim which I explained the textual reasons for on my Sola Scirptura blog. It most likely refers to Rhodes, an island north-east of Crete and closer to Turkey then it is to Crete, south of Ionia. KJV onlyists however will cling to Dodanim irrationally, which makes Dodona an attractive identification. Dodona was an Oracle in Epirus even more ancient then Delphi, and also a city in Thessaly. Aristotle said Dodona was the original homeland of the Hellenes, but history wasn't his area of expertise. Attempts to say they were actually the Dardanians are also made, but there is no textual support for a d-r reading.
Tarshish is who's identification is the most mysterious. I do not think possibly misunderstood Chronicles verses are good justification for placing Tarshish in the east rather then west. Regions of ancient Spain like Tartessos I think were first settled by Phut, there has long been speculation that the Basques are related to the Berbers. And if I were to theorize a Biblical origin for the name Tartessos itself, Tirzah is closer then Tarshish since there are reasons that letter for Z sometimes become a T.
I no longer support fanciful theories about Tarshish being Briton or India or Japan. However of mainstream theories Tarsus is the most probable.
Tarshish is another example of my maybe coming to support a theory of Velikovsky I originally didn't think I would, which is making them the Minoans/Crete or more specifically Knossos.
https://www.varchive.org/nldag/tarshish.htm
However it could also simply be the Tarsus of Cilicia which was already known by that name in Assyrian Inscriptions.
In Greek Mythology a people called the Telchines were the earliest inhabitants of Rhodes, and also lived on Cyprus and Crete. So this term could be a name given to the offshoots of Javan. And that makes them Pre-Hellenic not Hellens.
The Pre-Hellenic people of Greece were often called Pelasgians as a whole, yet Pelasgians are sometimes implied to be one specific group. Some theories about the etymology of Pelasgian imply there wasn't originally an S before the G, which makes a connection to Peleg possible.
I also believe two Canaanite tribes were among the Pre-Helelnic Greeks. The Arkite tribe were the Arcadians and the Sinite tribe the Sintians.
Thrace I think was the ancient nation of Tiras, but I think Thracians also traveled north and contributed to Scandinavia.
Where do I think the Hellens came from Biblically? Well I think they may have been Israelites who lost their identity. Partly from north western coastal tribes of the Northern Kingdom becoming sea traders and thus mingling with the Phoenicians as well as Ionians in cities like Miletus. And partly from what Joel 3 says about children of Judah and Jerusalem being sold as slaves to Javanites by Tyre and Philista. Which could include the wives and children of Jehoram of Judah some of whom were taken by the Philistines.
On my comparative mythology blog I shall in the future discus evidence for that from Greek mythology. But one particular factor there, since the Joel verse gives good reason to see Benjamites as included, is that the Tarsus of Cilicia could come from the Tarshish of the Tribe of Benjamin in 1 Chronicles 7:10. Paul was born in Tarsus of Cilicia and is well known to have been a Benjamite.
But for now I want to remind those trying to argue Paul only brought The Gospel to the "Lost Tribes" that specifically Ionian places are important in Acts. Ephesus and Miletus in Ionia (likewise Ephesus and Smyrna in Revelation), Athens, Cyrpus, Rhodes and Crete were all visited by Paul. And then of course Paul's hometown may itself be Tarshish.
Update October 2020: I'm updating this in the context of my Language of The Table of Nations discussion on another blog.
The Linguistic unity of all the subtribes of Greece has caused me somewhat change what I argued above and still view all of them as ultimately of Javan. First look at this Map from the Dorian Invasion Wikipedia page.
Elishah I believe was the Northwest Greeks and the Aeolians.
Further Update June 2021: There is plenty of reason to suspect the region of Ionia is where all the ancestors of the Hellens lived first. Linguistically that Phrygian is considered closely related to Greece fits, and the Anatolian Languages might be as well, perhaps they are the branch of Javan represented by Tarshish. Archeologists don't believe Hellenic speakers arrived on Western site of the Aegean till about 1600 BC. Revised Chronology could bring that down another 4 or 5 centuries.
Homer is believed to have lived on an Island on Anatolia's side of the Aegean, and Hesiod is said to have been born there as well. Ionia is also where Pre-Socratic Greek Philosophy started, the three Milesian Philosophers, Heraclitus of Ephesus, Xenophanes of Colophon, and Pythagoras was born in Samos.