The characters for Yagokoro literally mean "Eight minds", and Eirin "Eternal jewel". "Eight" is the sacred number especially for ancient Japanese, and it could mean "sacred, numerous, infinite". For example, Yakumo (八雲, eight clouds), Yashima (八島, eight islands), Yata no Kagami, Yaoyorozu no Kami. And gokoro is an ancient reading for 意, which could indicate something about heart or mind. So her whole name could mean "Intelligence of eternally clear brightness over infinitely extended area", referring her to be the "Mind of God".
Eirin's clothing has trigrams (broken and solid Yin/Yang lines) lining the bottom hem. They appear to be the complete set of trigrams used in Feng Shui, and match the set of trigrams on the Hakkero.
The Ghost Team's ending in Imperishable Night reveals that Eirin herself has consumed the Hourai Elixir (lost in translation, especially due to not being stated directly). The gist of the conversation is that she has taken medicine prepared through Kaguya's power and therefor can never become a resident of the Netherworld. Yuyuko is taken back, claiming it makes Eirin a "natural enemy." Along with Kaguya and Fujiwara no Mokou, she is among the few immune to Yuyuko's power of death.
Eirin is one of the few characters that revive (along with Yuyuko, Kaguya, Tenshi and Suwako) after being defeated the first time.
Lunarians have very long lifespans, but only by remaining free of impurity such as life on Earth. Escaping such impurities is why the the capital on the moon was created in the first place. Just as Reimu can negate it by introducing the tainted life of her home planet, the benefits are only in effect so long as the inhabitants stay in the lunar capital. Eirin would not have the benefit of her Lunarian lifespan on Earth. Choosing to follow her princess in eternal exile, she would have had to take the Hourai Elixir.
Inaba of the Moon and Inaba of the Earth contains a contradiction when Eirin becomes sick. This should be impossible for an immortal who has taken the Elixir. On the other hand, the series is a light-hearted comedy not written by ZUN and not exactly canon.
Eirin has a Spell Card named "Apollo 13" after the aborted third American moon mission launched in April 1970. The spell card's spell sign is "Curse of the Heavens", and seems to imply that it was Lunarian magic, not mechanical failure, that critically damaged the spacecraft and forced it to abandon its lunar landing so as to forestall the "invasion". It's also stated that the Lunarians know why the number 13 is cursed.
She might be based on Ame no Yagokoro Omoikane no Mikoto (天八意思兼命), the god who thought of the plan to lure Amaterasu out of hiding. This can be seen by her spell card, and by the correlation between Eirin's sharp intellect and Omoikane's. (Takami-Musubi is also very strongly associated with intellectual facilities.) The interrelationships of various lunar figures presents a more interesting situation. Eirin is distantly related to the Watatsuki sisters, both based on mythology. In Cage in Lunatic Runagate: Chapter 3, Watatsuki no Toyohime mentions that Eirin's participation was critical to the founding of the lunar capital, and that she was older even than Tsukuyomi. Omoikane is the son of Takami-Musubi, one of the great gods that summoned Izanagi and Izanami into existence, but it's unclear whether Omoikane was born before the creation of the Japanese islands, which also saw the births of Amaterasu, Tsukuyomi, and Susanoo. Regardless of whether she is "meant to be" Omoikane, Eirin clearly draws on Omoikane in many ways.
Eirin had recently developed a medicine called Kochoumugan (Butterfly Dream Pill), which guarantees the person who takes it will have a pleasant dream once they fall asleep. The medicine was featured in an article in the Bunbunmaru Newspaper, but still remains fairly unpopular (mostly due to the fact that the paper itself isn't too popular). There is one customer, Alice Margatroid, who has returned numerous times to purchase the medicine however, although the reasons why were never disclosed.
In Tibetan Buddhism, the bow held in the right hand with an arrow in the left (as Eirin holds them) is a symbol of wisdom as a means of projecting method or skillful means.
The bow and the arrow are a possible reference to the huntress greek moon goddess Artemis (sister of Apollo; In Imperishable Night, ZUN made a joke about Apollo being the sun god, so he surely knows about Artemis). Artemis (or her bow) sometimes represents the crescent moon, symbol of full maturity in the cycle of life (it suits Eirin very well).
Artemis is often assimilated to Selene (the personification of the moon), which is the root of Selenium, a non-metal antioxidant sometimes used as a medicine but toxic in large quantity to human beings. Antioxidants are known to improve life expectancy.
Ursa Major, one of the constellations on Eirin's clothes, was originally Callisto, a servant of Artemis.