Green’s Dictionary of Slang

(US) a martini cocktail.

1951		R. Starnes Grant’s Tomb 212: ‘May I order you anither [sic] martooni? he asked.
1960, 1975		Wentworth & Flexner DAS.
1974		E.H. Hunt Undercover 94: [H]e ordered a round of double ‘martoonis’ for all of us.
1983		S. King Christine 424: She didn’t like to think of her dad trying to drive home [...] half-soused from three or four martinis (except that he always called them martoonis, with typical adult kittenish).
1996		N.Y. Observer 22 July 34: Pack up your picnic basket of martoonis and head your [...] Saab to Jersey [HDAS].