The characters in her name literally mean Hi or "compare, comparative, rather", the former na is "many, comfort, calm", the latter na "name, authority", i "be, live", and ten "sky, heaven", and shi, which is a common ending for girl's names meaning "child". So the literal meaning of her name is "The child of heaven, from the calm family which bears authority".
But there is another possible meaning. Her name includes hina and nai. The meaning of Hina isn't clear, but the official profile says the Hinanai clan served the Nai clan in their life, so hina could mean "young chicken, small" (雛). In ancient times, na could have meant "ground", i or wi could have meant "be, exist, live in", the compound nai or nawi "foundation, basement" (the ground on which we stand), and naifuru was an ancient word for "to have a earthquake" (for the ground on which we stand to be shaken), which you can find on Nihon Shoki. As time passed, people changed "naifuru" to "for an earthquake to occur", then "nai" came to mean simply "earthquake". So her name means "The child of heaven, from the small family serving the Nai clan, who are in charge of earthquakes".
Her profile mentions "Oomura no Kami" and "Nai clan". In fact, there is the Nai Shrine (名居神社) in Nabari City, and the Oomura Shrine (大村神社) in Iga City, both in Mie Prefecture. The former shrine stands in Shimo-hinachi (下比奈知), and the latter has the Keystone and the Statue of Catfish as goshintai.
Sometimes Tenshi's first name can also be read as "Tenko": when the fans were speculating about new charcters on Touhou Project Board before the release of Scarlet Weather Rhapsody, an anonymous poster jokingly uploaded an image that supposedly listed all the names of the characters including that of an unknown character, "緋想 天子" (engl. Tenko Hisou) in the middle. This name was too obvious; adding the most common Japanese suffix for girls' names, "-ko", after the title "Hisouten". The name was from then on adopted as a tentative name for the final boss, because such an obvious name sounded very funny. Surprisingly, it turned out that she has those same characters in her first name, "天子", but a different pronunciation, "Tenshi".
As a result of this, Tenko has been a fan nickname for her, and she sometimes participates in Ran Yakumo's Suppa Tenko running gag.
Tenshi when written as "天子" can also mean "the emperor", being the son of heaven.
Tenshi in Japanese can also mean "angel" (天使), a likely reference to her being a Celestial, and an ironic use of the word (considering Tenshi's actual character) by ZUN, in line with the other lampooning of Heaven and Heavenly beings within Touhou.
While Tenshi is called a Tennin (天人) many of the mythological references Tenshi makes are those of the Tenbu (天部, Deva in sanskrit) In Buddhism, Deva are humans that have achieved enlightment and in Scarlet Wheather Rhapsody, Tenshi mentions the five signs of death of a Deva. Deva's are also known to be mortal in Buddhism, living long lives, but still remaining part of the circle of life and death. This too, ties in with the events in Komachi's SWR scenario. Considering earlier references by ZUN to the Tennin (such as several of Youmu's spellcards in PCB) both Tenbu and Tennin are the same in Gensokyo.
One of her color palettes in Hitsoutensoku resembles Renko.
Tenshi is also one of the characters to have revival spells.
Tenshi has appeared on the official shirts of the Anime Weekend Atlanta 2013 with ZUN's signature, where he was a guest of honor.