"flat of the hand," c. 1300, from Old French palme (Modern French paume), from Latin palma "palm of the hand," also "flat end of an oar; palm tree," from PIE root *pele- (2) "flat; to spread" (source also of Greek palame "open hand," Old Irish lam, Welsh llaw, Old English folm, Old High German folma "hand," Sanskrit panih "hand, hoof").
Palm oil is earlier in the punning sense of "bribe" (1620s) than in the literal sense of "oil from the fruit of the West African palm" (1705, from palm (n.2)).
The word traveled early to northern Europe, where the tree does not grow, via Christianity, and took root in the local languages (such as Old Saxon palma, Old High German palma, Old Norse palmr). Palm Sunday is Old English palm-sunnandæg.
In ancient times, a leaf or frond was carried or worn as a symbol of victory or triumph, or on feast days; hence figurative use of palm for "victory, triumph" (late 14c.). Palm court "large room in a hotel, etc., usually decorated with potted palms" first recorded 1908.