The story
Unknown etymology, 1813. Original sense “alcoholic drink”, possibly a variant of switchel (“a drink of molasses and water, often mixed with rum”), attested 1790, itself of uncertain origin. Possibly influenced by swirl or fizz. In verb sense “to stir”, from swizzle stick (“stick for stirring alcoholic drinks”), itself attested 1859.
Etymology from Wiktionary, CC BY-SA 4.0.