The term “Dzenodoo” appears on page 173 of The Secret Doctrine, Vol. 1. In the original 1888 edition:
The seven “mysteries” are called by the Japanese Yama-boosis, the mystics of the Lao-Tze sect and the ascetic monks of Kioto, the Dzenodoo—the “seven jewels.”
HPB first mentions this sect in “A Bewitched Life, As Narrated by a Quill Pen,” Theosophist, August-September, 1885.
He belonged to the temple of Tzi-onene, a Buddhist monastery as famous throughout Tibet and China as in all Japan. None other is so venerated in Kioto. Its monks belong to the sect of Dzeno-doo, and are considered as the most learned among the many erudite fraternities. They are, moreover, closely connected, and allied with the Yamaboosi, (the ascetics, or “hermits”) who follow the doctrines of Lao-tze.
Tzi-onene = Chion-in (知恩院, chi on’in), a major temple in Kyoto.
The sect of that monastery is the Jōdo-shū (浄土宗), or just Jōdo (浄土), since shū simply means “temple” or “school.” This is a sect of the Pureland Buddhists in Japan, and Jō-do literally means Pure-land.
Now, 浄土 is Japanese, but comes out of the Chinese for Pure-land. 净 is the Japanese variant of the Chinese 淨, so in Chinese the term is 淨土 (Jìng-tǔ, Pure-land). Now, if we examine the Chinese characters:
浄 = jìng (pure). In middle-Chinese the pronunciation was /d͡ziᴇŋH/, i.e. “dzen-u.” The “u” sound is voiceless, meaning that instead of ending with the “n” sound, you follow through with a kind of “u” in the throat without really voicing a vowel sound. This is common across several languages, where instead of ending with a hard stop on the final consonant one will finish with a subtle “uh” at the end.
土 = tǔ (land); in middle-Chinese the pronunciation was /tʰuoX/, i.e. “tuoo” or “too.”
So, in middle-Chinese, the pronunciation would be roughly “dzen-u-too.” Blavatsky’s spelling is “Dzenodoo.”
We can do the same exercise with the temple name above:
知恩院 when in Japanese is chi on’in, but in Chinese it is zhī-ēn-yuàn (in modern pinyin). In middle-Chinese, the first character 知 is pronounced “ti” (/ʈiᴇ/) or “tzi”; the second character 恩 is pronounced “an” or “on” (/ʔən/). The final character 院 also has that same voiceless ending (/ɦˠiuᴇnH/), thus instead of “yuan” it is more like “heeon-u.” If we put all that together, we get a pronunciation something like “tzi-on-een-u.” Blavatsky’s spelling is Tzi-onene (i.e. tzi-on-en-e).
Quite likely you’d find similar pronunciations of these terms in different Chinese dialects. Given that Blavatsky’s experience was primarily with Tibetans and Kashmiris, as opposed to native Chinese and Japanese speakers, it is quite possible that the pronunciation she heard was from non-native speakers. It is also possible that Blavatsky’s teachers used middle and old Chinese pronunciations overall, as seems to be the case with the term Dzyan, for instance.
In conclusion: the Dzeno-doo sect of the Tzi-onene temple is the 浄土 (Jōdo) sect of the 知恩院 (Chion-in) temple.