Ironically the closest living relatives to the sea people that settled there are probably the Ashkenazi.

Back in 2018, a user on a genetics forum had strange result when they looked for the best match for Ashkenazi DNA. It was a recent sample from 3,500 year old graves in Crete taken for this research.

It led to 10,000 pages of discussion.

One of the things that was odd in that discussion was it seemed not to have much of the Doric DNA that entered Greek lineages around the time of the Mycenaeans in 1300 BCE, and was instead closer to Bronze Age Minoans and Anatolians. Which makes no sense if this was a 900 CE admixture from around the time of the bottleneck emergence of the Ashkenazi in Europe.

There's since been published research commenting on the high correlation between Cretean and Ashkenazi DNA:

Furthermore, we find in both PCA and ADMIXTURE analysis, that the Ashkenazi are more similar to the Cretans than to the two Levantine Semitic populations. One possible explanation is that this relation might reveal a common Mediterranean ancestry that the Cretan and Ashkenazi populations share.

But it gets weirder.

In some of the earliest Ashkenazi remains in Europe that have been analyzed researchers found the G2019S variant on LRRK2. This is a mutation associated with increased Parkinson's risk and is found in about 20% of Ashkenazi with Parkinson's. Few other populations have it at this frequency, but one population has it even more often - among the North African Berbers 40% with Parkinson's have it.

In fact, a 2017 study into the mutation concluded it originated among the North African Berbers, but found a very puzzling detail regarding its presence among the Ashkenazi:

However, a problem arises when we attempt to explain the high frequency of this mutation in the Ashkenazim population. The G2019S mutation in Ashkenazim was reported to arise 4550 years (3250–6425) years ago [11] using a multi-ethnic ancestral haplotype. This age estimation, being slightly younger than that of our Berber ethnic group, is prior to the beginning of Jewish Diaspora and its establishment as an ethnic Jewish group.

Taking these two oddities together, of high overlap of Ashkenazi DNA with pre-Doric Aegean/Anatolian DNA and the presence of a North African mutation that overlaps with a possible late Bronze Age admixture and an elegant solution comes into focus.

As you might know if you've ever looked into the sea peoples, their earliest mention in connection with the sea was in Merneptah's Libyan War inscriptions where they were allied with the Libyan Berbers against Egypt around 1200 BCE. The sea peoples there were described as being without foreskins, and at least one of the tribes overlaps with the 12 groups of tribes Ramses II had brought into captivity following the battle of Kadesh (one for each son with him).

What's interesting given all of this was a comment made by Tacitus that's generally dismissed by modern historians:

It is said that the Jews were originally exiles from the island of Crete who settled in the farthest parts of Libya [...]

This is dismissed because we know pretty much for sure that the Israelites emerged from the local Canaanite population with no migration or Exodus around the LBA/Early Iron Age.

But Ramses III alleged he forcibly relocated the sea peoples into the Southern Levant, we know an Aegean or Anatolian population conquered Ashkelon in the early Iron Age and had kids with the local population, and there's been a number of recent finds of Aegean style pottery made with local clay from the LBA/Early Iron Age in various sites in the Southern Levant, including Tel Dan leading the lead site researcher to think there's credibility to an old theory that Dan were actually the Denyen sea peoples also found up in Adana in Anatolia.

Maybe there was a continuous subpopulation among the Israelites tracing back to a LBA/Early Iron Age mixture of Libyan and Aegean/Anatolian sea peoples?

We actually can see a hint of this in the Bible, with Lamentations 4:7 discussing the Nazirites (a population whose vows involved killing an entirely red haired cow and who couldn't cut their hair) pre-Babylonian exile as being pale skinned and described in an honored way, but then 2 Kings 5:27 describes a pale skinned population whose children are also pale skinned as being the result of a curse from God. One of the dead sea scrolls even strangely claimed Noah had red hair (like the North African Berbers).

In fact, the alleged reforms of Josiah (anachronistic that early given letters between Elephantine and Jerusalem) are mostly positioned as opposing the traditions of Jeroboam, the figure who allegedly had all the tribes except Judah and Benjamin behind him, and whose either grandmother or mother is simply identified as 'leper.'

Maybe a continuous endogamous and matrilineal subpopulation from the sea peoples present in the Southern Levant goes from a respected position early on to an increasingly marginalized one until eventually after the first temple falls they gradually made their way up across Europe.

There's a lot more interesting historical context to the events taking place from around from the 13th century BCE to the 8th century BCE, but this comment is long enough. For those interested, I discussed a lot of those elements in a thread that starts here in /r/AcademicBiblical on Reddit.

TL;DR: Maybe the half-descendents of the sea peoples are still around and many are back in that area today.

Though if we're discussing the notion of indigenous claims, the most ancient population for the area were the Canaanites, who are the core predominant ancestors of both the Palestinians and Jews. i.e. In an ideal world we'd be removing the religious orthodoxy from the picture in dividing what's essentially a singular group of closely related ancestry against themselves.