The word is made of 2 kanji:*
霞 kasumi fog, mist, haze
関 seki barrier (gate)
There are competing theories about this place name.
1) It was the place where the land that separates clouds and fog from the solid land.
2) It was a place where you could look down and view the clouds or mist.
3) There was a 関所 sekisho highway-checkpoint here on a road that allegedly became the 奥州街道 Ōshū Kaidō Ōshū Highway. There is a reference to a 霞ﾉ関 kasumi no seki in the 1360’s in the region.
4) It was the boundary that separated the Yamato people from the “Eastern Barbarians” in the “Eastern Country” by the mythological 12th emperor 日本武尊 Yamato Takeru. Thus implying that on the other side of the 関 seki border, there was 雲霞 unka clouds & haze, barbarianism, (barbarian) swarms.
There is another 霞ヶ関 Kasumigaseki in Kawagoe, Saitama. The place names may be related, but we can’t say definitely. In my personal opinion, all four sound a little fishy.
1) All land is separated from clouds – why would you need a name a special place for that?
2) Kasumigeki isn’t so elevated that you could look down on clouds or fog; and even if you could, it would still be foggy there too so you couldn’t see anything.
3) The presence of a military checkpoint on the road is not unimaginable, but it doesn’t really explain the kanji 霞 fog.
4) The Yamato Takeru thing is the most difficult to prove or disprove since we’re dealing with a mythological Emperor. But by the time the place name got written down on maps that we still have today, there would be no way to confirm or disprove the claim. And even at that, it still sounds made up to me. Come on, the dude’s name was 日本武尊, we might as well translate it as “Captain Japan.”
Well, as long as we’re just throwing out random theories, I’m gonna throw some of mine out there. Anyone can play this game!
The kanji 霞 appears in some other words, 霞草 kasumi sō (baby’s breath, a flower), 霞石 kasumi ishi (nepheline, a mineral), 霞桜 kasumi sakura (Korean hill cherry, a tree). Any of these plants or stones might have been in the area. Also, if the name predates the Edo Period, there is always the possibility of it being a local name which may have origins in the local dialects (which disappeared during the Edo Period or even later by the creation of 標準語 Hyōjungo Standard Japanese in the Meiji Era). Sadly, I fear we may never know the true origin of this place name.
* The middle character ヶ, read “ga” is shorthand for a classical Japanese genitive particle. An alternate particle can be rendered as 乃 and 之 (no) and in modern Japanese by の, ノ (no).