Based on the accepted answer, here's the transcription of the stages and some considerations by a player of the game.
They are named by the stars of the Big Dipper. From Wikipedia:
The Big Dipper (US, Canada) or the Plough (UK, Ireland) is a large
asterism consisting of seven bright stars of the constellation Ursa
EDIT: As @BlueRaja - Danny Pflughoeft said below these aren't accurate English translations:
Unfortunately most/all of the translations given by google are just
straight up wrong, because they're not actual words in Japanese (or
Chinese, as far as I can find). For example, 廉貞 ("pure" +
"rightousness") is translated as "innocent", but that's just a total
shot in the dark by Google. And 文曲 ("sentence" + "music") is
translated as "bunka", which is not even an English word (it's
Google's best guess for how that it would be pronounced, if it were a
With that said, my guesses of the meaning of these names in the game are even more abstract.
Ursa Major η star (Alkaid)
- Stage 1 A: 揺光 - Shaking light
- Stage 1 B: 破軍 - Defeated army
'Defeated army' or 'Broken Troops' may refer to the Oboro Clan in the game.
Ursa Major ζ star (Mizar)
- Stage 2 A: 開陽 - Kaiyo
- Stage 2 B: 武曲 - Martial arts
Ursa Major 3 star (Alioth)
- Stage 3 A: 玉衡 - Tamaki
- Stage 3 B: 廉貞 - Innocent
Innocent may refer to the Hiruko's niece in the game.
Ursa Major δ (delta) star (Megrez)
- Stage 4 A: 天権 - Heaven
- Stage 4 B: 文曲 - Bunka
Ursa Major γ star (Phecda)
- Stage 5 A: 天璣 - Tenri
- Stage 5 B: 禄存 - Resident
Ursa Major β star (Merak)
- Stage 6 A: 天璇 - Tenryu
- Stage 6 B: 巨門 - Giant Gate
Ursa Major α star (Dubhe)
- Stage 7 A: 天枢 - Heavenly center
- Stage 7 B: 貪狼 - Greedy wolf
Heavenly center may refer to the place where Stage 7 is, certainly very special for the characters of the game.
Greedy wolf may refer to the Akujiki Sword or even the 7-B's boss
Ursa Minor α (North Star)
- Stage 8 A: 北辰 - Hokushin
- Stage 8 B: 太一 - Taichi
So each stage refers to one star. "A" refers to their Chinese names and "B" to their names in Buddhism.
At the beginning of the Stage 8, there's a cutscene where the last boss talks about the 7 stars and their counterparts:
On Wikipedia you can also see the constellation as the boss shows, a little different, but the idea is the same.
You suppose to start by Alkaid and go all until Dubhe, and then reach the North Star (that is not in Ursa Major, but is the brightest of the Dipper), stage 8, the Golden Palace, that in the game is a place emerged from hell, the world of the dead, or something, and is placed on the center of Tokyo.
I don't know much about Asian religions. They could get a little dipper, I mean, deeper into this, the back story of the game tells almost nothing about the stars.