The story
From wet (“to celebrate by drinking alcohol”) + the + shamrock (“trefoil leaf of any small clover, especially Trifolium repens, or a clover-like plant, commonly used as a symbol of Ireland”), from the custom, also known as drowning the shamrock, of removing a shamrock worn on one’s clothing on Saint Patrick’s Day and placing it at the bottom of a glass which is then filled with an alcoholic beverage and consumed; according to some authorities the shamrock is then retrieved from the empty glass and thrown over the left shoulder.
Etymology from Wiktionary, CC BY-SA 4.0.