So as both an Egyptian and a historian of empire, I hope I can shed some light on this common misconception.

Modern Egypt is actually quite clearly descended from Ancient Egypt culturally as well as genetically. Coptic, the liturgical language of Egyptian Christians, is the [latest form] (http://glottolog.org/resource/languoid/id/copt1239) of the ancient Egyptian language. The [Coptic calendar] (http://st-takla.org/Full-Free-Coptic-Books/Coptic-Synaxarium-or-Synaxarion_English/Eng-Synexarium-or-Synexarion-index.html) is also derived from the ancient agricultural calendar. The spring harvest festival Sham el-Nessim traces its history back far enough to have been described by Plutarch around 46 AD and is still a widely celebrated holiday, probably the longest running one in the world. Egyptian Arabic contains pharaonic and Coptic words; even the universal sheb-seb (flip flop) comes from the Coptic word for foot. There are also some grammatical quirks that sound strange to other Arabic speakers that come straight from Coptic. (I can't find a specific source for these two points right now but I'll check some Arabic resources when I get home.) Even geographically, Egypt has had fairly consistent borders for much of its history prior to the nation-state, thanks to the Nile making a very specific area habitable; ancient Egypt also encompassed modern Sudan.

Genetically, there have been [some changes] (http://www.sciencemag.org/news/2017/05/scientists-thought-ancient-egyptian-mummies-didn-t-have-any-dna-left-they-were-wrong); most notably, modern Egyptians contain much more sub-Saharan African DNA, while both ancient and modern Egyptians closely resemble ancient Levantine and North African peoples. In addition, most modern Egyptians do possess genetic markers going back to North and Northeast Africa. (The History and Geography of Human Genetics is a good, albeit old, resource about this. I also refer to [these] (https://www.nature.com/articles/srep09996) [studies] (http://muse.jhu.edu/article/14056))

Ok, now some notes on our Arabness. The Arabs were not colonizers who changed our ethnicity but rather strategic and military conquerers. It did not implant culture in the imperialist sense, although it did bring religion. The adoption of Arabic was slow and gradual and went hand-in-hand with the conversion to Islam. Modern Arabization was a very recent (like, 1950s) project undertaken by Abd el Nasser primarily as a self-determination strategy. The self-perception of Arabs as a homogenous group was a direct response to independence from Britain and France throughout the Middle East, with Egypt at the forefront. It resulted in the suppression of some local cultures, particularly the Nubians of Upper Egypt, but it was an assertion of identity and a political choice rather than an absolute truth. Egyptians do not struggle with this. Egyptianness/pharoanicness all coexist fine with Arabness for every Egyptian I know. (Sacred Language, Ordinary People: Dilemmas of Culture and Politics in Egypt and The Origins of Arab Nationalism are useful here.) Even now, el Sisi has been trying to play up Egypt's Africanness and improve relations with the rest of the continent, which very much went downhill due to the exploitation of the Nubians under el Nasser and some stuff Morsi did to cause issues pretty much the whole continent.

This coexistence is possible because Arab is not an ethnic label, it's a cultural and linguistic one, despite what many Westerners believe. It's a fluid and changing thing. Most Egyptians I know would identify as Egyptian first and Arab second; many Copts especially don't call themselves Arab at all but rather thouroughly Egyptian/pharaonic. None of this is meant to deny that Egyptians don't have issues with colorism and racism, but those are the result of the effort to apply racial constructs to a MENA context, where they almost never sense (my dad, for example, is typically read as Black, while I'm never read as anything but white). I also don't mean to imply that Egyptians haven't changed genetically or phenotypically over time; every group has. It is simply meant to argue that Egyptian ethnicity has remained intact. We're still called Egypt because we are still the home of the Egyptians. Our culture and genetics have changed because that's what they do.

What's the moral of the story? Modern Egyptians are the most Egyptian people that exist and absolutely have a claim to their own history. I hope this helps! I tried to cite sources but I'm on my phone so let me know if you want any for specific points or have any follow-up questions.

edited for weird phrasing

Could anyone expand on Khemet or Khemetic culture? I've heard that this is the name 'Egyptians' used to refer to themselves.

I only know about Egypt and Greece.

Egypt is called like that because that's how Greeks initially called it, and later on Romans adopted the name, and through the Roman Empire, the west came to call this country Egypt. As for the origin of the name, it is given in a Greek mythological genealogy of the human tribes. The name originates from Egyptos, the brother of Danaos.

As for Greece, it is kind of more complicated. The Greeks were an indigenous tribe of the place we call Greece. Ancient Greeks(in general, not the tribe) though didn't collectively called themselves Greeks, but Hellenes. Greeks again were called like that after the 'forefather', Graecus, who was said to be a son of Deucalion. Ancient Greeks believed that the 'Greeks' were the most ancient tribe / nation of the world, and thus the name in Greek is translated as "Oldfolk". Because the ancient Hellenic tribe of the Greeks spoke the same language as the rest of the Hellenic peoples, and had the same religion, the Romans called collectively Hellenic peoples "Greeks". And through the Roman Empire, the name was spread in the West. Similarly the first Hellenic tribe the East met when Alexander the Great occupied the Eastern countries, were the Ionians, and as a result most Eastern countries call Greece to this day the equivalent of 'Ionia'.