In 1586 Sir Francis Drake sailed into to Havana Cuba. He was sent on a mission by the British Crown to the Spanish colonies in the Caribbean to plunder their gold. The reason he came to Cuba was another. His crew was suffering from scurvy and dysentery and they needed to find a remedy. The locals brought him Aguardiente (an early form of rum), lime, mint and cane juice, a concoction they started to call El Draque, the name they gave Sir Francis Drake. The drink did not however cure himself and Sir Francis Drake died of dysentery outside Panama in 1596.
The later name Mojito might have derived from the Cuban mojo, a sour citrus sauce made by African slaves.
This blend of alcohol, lime, mint and cane juice was most likely drunk by slaves on the Caribbean plantations to help with their own ailments long before Drake came to try it. But not until the American Prohibition when the American cocktail party moved to Cuba the drink started being served as a highball with soda and the Mojito as we now know it was born.
The highball glass was designed by Finnish glass designer Nanny Still in 1964 and is called Flindari.