Her first name, "Patchouli," could be taken from name of a herb used in perfumes and incense. Her last name, "Knowledge," appears to be a reference to her amount of knowledge, though few other characters seem to have such literal surnames.
The actual Japanese pronunciation of her first name goes more along the lines of "Pa-chuu~ri" as is written out by the katakana, which is 「パチュリー」 ("Pachurii~") rather than 「パチョリ」 ("Pachori.") Thus, her abbreviated nickname is 「パチュ」 or "Pachu~," and more commonly, 「パチェ」 or "Pache-" (pronounced "pa-chae").
After being defeated in Marisa's scenario in IaMP, she utters "mukyu~". Since then, fan works have often depicted her using this sound as a sign of surprise or pouting.
The crescent on Patchouli's hat could be the crescent in the Sanskrit "Hum", part of the Mantra "Om mani padme hum". In Buddhism, it stands for knowledge and purity and is associated with the color white—all the 7 colors combined.
She has waterproof books that Marisa finds interest in reading in the bath.
Patchouli was born with asthma, for approximately 100 years.
Patchouli is the only character so far to have a medical condition.
As of Double Spoiler, Patchouli has had appearances in 9 of the 14 windows games (counting Hisoutensoku as a standalone). This makes her the third most featured character, behind Reimu and Marisa.
Patchouli's magic based on elements of Wood, Fire, Metal, Water, and Earth. In Chinese signs of the Zodiac there are five beasts: Azure Dragon, Red Bird, White Tiger, Black Turtle and Yellow Dragon. Each of the five beasts refer to same elements Patchouli use: Azure Dragon - Wood, Red Bird - Fire, White Tiger - Metal, Black Turtle - Water and Yellow Dragon - Earth.
Patchouli's weakness and ill health is caused by the build up of arsenic and mercury, which she has to use a lot in magic spells. This is stated in Wild and Horned Hermit Chapter 3, Page 23.