The story
After French mathematician Siméon Denis Poisson (1781-1840), who introduced the distribution in his 1837 work Recherches sur la probabilité des jugements en matière criminelle et en matière civile ("Research on the Probability of Judgements in Criminal and Civil Matters"), although prior work had been done by Abraham de Moivre.
Etymology from Wiktionary, CC BY-SA 4.0.