"A small gillywater --"
-- Madam Rosmerta serving Minerva McGonagall a drink at the Three Broomsticks (PA10)
A drink served at the Three Broomsticks in Hogsmeade.
Minerva McGonagall ordered this drink from Madam Rosmerta (PA10).
Gillywater may be made from gillyweed, a plant that when eaten helps someone grow gills to breathe underwater. Harry used the plant to help him swim in the Hogwarts' Lake during the Second Task of the Triwizard Tournament (GF26).
Old Norse gjölnar which perhaps means "fish gills"
"gilly" from "gillyflower" = any of several species of fragrant small flowers often carried or displayed in a small bunch.
This word is similar to "Gigglewater," an alcoholic beverage served in the Blind Pig, a New York speakeasy in the 1920s.
It seems fitting that the Transfiguration teacher would enjoy a drink made from a plant which transfigures people.
Minerva ordering gillywater seems like foreshadowing for the way Harry is told about gillyweed in the following book. Fake Moody, Barty Crouch, Jr., wanted Harry to win the Second Task of the Triwizard Tournement, so he gave Neville Longbottom a book about water plants (GF35). Harry never used the book, so Barty asked Dobby to clean the Staff Room, then staged a conversation with Minerva about gillyweed so the House Elf would overhear. Of course the ever-loyal Dobby went directly to Snape's office and stole gillyweed to help Harry win (GF26).