1680s, pimiento (modern form from 1718), "dried, aromatic berries of an evergreen tree native to the West Indies," cultivated mostly in Jamaica, from Spanish pimiento "pepper-plant, green or red pepper," also pimienta "black pepper," from Late Latin pigmenta, plural of pigmentum "vegetable juice," from Latin pigmentum "pigment" (see pigment (n.)). So called because it added a dash of color to food or drink.
[I]n med.L. spiced drink, hence spice, pepper (generally), Sp. pimiento, Fr. piment are applied to Cayenne or Guinea pepper, capsicum; in Eng. the name has passed to allspice or Jamaica pepper. [OED]
The tree itself so called by 1756. The piece of red sweet pepper stuffed in a pitted olive so called from 1918, earlier pimiento (1901), from Spanish. French piment is from Spanish.