An aqueous martini is an alcohol-free, water-based variant of a martini cocktail. It consists merely of cold water with an olive on top, served in a fancy glass.
At the time that Violet, Klaus and Sunny Baudelaire came to live with Jerome and Esmé Squalor, aqueous martinis were the "in" beverage. As they tasted of nothing but plain water with just a hint of olive, Violet and Klaus did not particularly care for aqueous martinis; Violet did not like the taste much but still found them thirst-quenching, while Klaus mentions in The Slippery Slope that drinking aqueous martinis put him off the taste of olives. However, aqueous martinis quickly went "out", to be replaced by parsley soda, which was even more unpleasant.