"Gengetsu" means "illusory moon" when the kanji are read separately; read together they mean "paraselene" or "moondog", which is a type of optical illusion.
The title of Lotus Land Story's Extra Stage is "Puckish Angel", and the white wings that she has look like angel's wings... For a demon, she's quite angelic...
The final events of Lotus Land Story occurred in a "Dream World." The place Mugetsu and Gengetsu reside in appears to be yet another dream world.
Her boss theme contains a short part of Yuuka's Stage 5 boss theme, Sleeping Terror. Also, the main melody is quite similar to Sakuya's Stage 5 boss theme, Luna Dial. However, it is quite common for Touhou songs to have allusions to other songs.
Gengetsu is able to fire a laser similar in appearance to Marisa's Master Spark.
The only difference from Master Spark is that while shooting, there would be lines of bullets and Gengetsu will be also shooting tiny bullets too.
The optional timeout phase of her last attack pattern has been given the nickname "Gengetsu Rape Time" by western players because of its extremely fast and hard to dodge bullets.