As the older sister, she is commonly called 「お嬢様」 ("ojou-sama") by fans, and is what is primarily used to refer to her. Whenever Flandre refers to her sister, though, she is called 「お姉様」 ("onee-sama"). Another common nickname is 「れみりゃ」 or "Remirya."
There is a common joke that Sakuya calls her 「おぜうさま」 ("ozeu-sama") instead, which has made it a very common nickname for her. The old pronunciation of 「お嬢様」 is "ojiyau-sama" (おぢやうさま) but in the Edo period, "jou" was colloquially written as "zeu," like "dozeu" etc.
In fanworks, whenever Remilia becomes embarrassed or humiliated, it is often referred to as a "Charisma Break" by the fanbase due to this being a contrast to her very charismatic personality.
A common meme regarding Remilia is 「れみりあ！うー☆」 ("Remilia! Uu-☆") which people think is cute.
In fanworks, Remilia is sometimes illustrated with Satori Komeiji, because they are both older sisters, and both are weaker than the younger sisters.