Taking a similar question, how much genetic drift occurred during the aftermath of the Muslim conquests between conquered areas and the peninsula itself?

Let me start this answer by stating that the Arab-Israeli conflict, as well as its history and origins, are perhaps the most politicized conflict in our modern history. There are so many opinions seeped into historical texts than it will be impossible to discern 100% objective fact about anything until the conflict is long over.

With that being said I will do my best to elaborate on the history of the ethnic backgrounds of Israel, but keep in mind that most of my knowledge pertains to the 20th Century. My answer here will draw on the knowledge base that 20th Century Israel is built on, as well as some research I've conducted in the last few days. Since we also don't have access to undisputed geological evidence (which I imagine that neither Israelis or Palestinians as ethnic groups have collaborative records of) we are going to assume that the populations of Jews and Arabs living in the area, despite frequent changes of power, were never changed in such a way that they are completely indistinguishable from the respective ethnic groups of today. I will mention whenever a change of power influenced the change of identity and culture, but I will leave the interpretation up to you. From here on I'll use the term Palestine to refer to the historical land of Israel and the Levant, the term Palestinian to refer to the predominantly Arab population in Palestine, and the term Israeli to refer to the predominantly Jewish population of Palestine.

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On your first claim:

language claiming that the palestinians are either very recent residents,

This is untrue depending on the exact definition of 'recent'. If recent defines any time period before the last 100 years, which I would argue is a very lenient definition of recent, the claim is simply unfounded. Palestinians have been a majority in Palestine for at least 1300 years when the Rashidun Caliphate invaded and occupied the land. While I'm sure there are Palestinians who have immigrated to Palestine since the formation of the State of Israel, during the most important time period before it's actual formation (1882 to 1948, when millions of Jews immigrated with support from international Zionist groups) there was exceptionally more immigration of Israelis than to Palestinians; thus there were more Palestinians of this time that were considered natives to the land.

The second claim:

or moved in only after the arab/muslim invasions of the 7th century

This claim is more truthful, but not entirely correct. Ancient Assyrian sources describe Palestinians living in the Fertile Crescent (which encompasses Palestine) as early as the 9th Century BCE. At this time its hypothesized that the Kingdom of Israel, the first ever time Israelis ruled independently as a people, held dominant authority over the land. However, the existence of this united kingdom is heavily disputed, because there are very few sources besides those found in the Bible that mention it. The same Assyrian sources attribute Palestinians living in Southern Palestine as early as the 4th Century BCE, as evident by the prevalence of Arabic names of inhabitants and people during the era when the Achaemenid Empire ruled over the area.

In between the 9th Century BCE and the 4th Century BCE here Palestine was conquered first by the Babylonians and then the Achaemenids. Both of these invasions had the potential to change the ethnic face of both cultures. And up until the Muslim invasions of the 7th Century Palestine was conquered by the Macedonians and the Romans, which also had the potential to change the ethnic face of both cultures.

Finally, in the 7th Century AD the Rashidun Caliphate invaded, which means that the second claim is partially true: the caliphate encouraged immigration to the region until eventually the majority of the inhabitants were Muslim. This is perhaps the earliest time that many historians agree that the Palestine was predominantly Arab, or more plainly, that the Palestinians population was a majority of the population of Palestine.

The third claim:

say that there used to be palestinian ethnic peoples who lived there but the modern Palestinians are not related to them

Unfortunately, as I mentioned above, the only way to confirm or deny this is to find genealogical evidence for the Palestinians and the Israelis and compare them, which I do not believe is even possible. I highly doubt that such evidence even exists. As such this claim is untrue.

On the last claim:

or simply try to argue that the Jews have been there longer.

Similar to the third claim, and as I mentioned with the disputed authenticity of the Biblical descriptions of the United Kingdom of Israel, there is simply not enough evidence to confirm this. Even by stretching the idea of successive invasions of Palestine not changing the ethnic identity of either culture, or that they weren't inherently united in the first place in pre-Biblical times, there is simply no way to prove Israeli identity was unique enough for this long to claim that they were there first. Thus, this is untrue.

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Thus, in conclusion, while one of the claims you have heard might be partially true, the others are either outright untrue or so vague that the evidence to support them has been lost to history. This isn't even taking into account the discussion that must be had to discern where the modern Israeli identity originated, or if the Palestinian identity had relevance before the Zionist Movement in the 20th Century. I've found an interesting description of DNA of Arabs who lived in Palestine on Wikipedia, but I don't know enough about it to elaborate or use it as a source. Hence, I recommend you read it yourself to see if it helps:

https://en.wikipedia.org/wiki/Palestinians#DNA_and_genetic_studies

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Sources:

• Eph`al I (1984). The Ancient Arabs, Magnes Press, Hebrew University of Jerusalem

• Hagith Sivan. Palestine in Late Antiquity, Oxford University Press, 2008, p.267, n.116.

• "Umar (634–644)", The Islamic World to 1600 Multimedia History Tutorials by the Applied History Group, University of Calgary.

• Smith, Charles D. The Arab-Israeli Conflict: A History with Documents, Bedford/St. Martin's Press, 2013, pg. 2-8.

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Edit:

As there was in fact immigration as pointed out by /u/Trancestor, I have edited my answer to accommodate this fact.