Everyone assumes the flower was named after the boy. The evidence does not back that up.
In Greek myth, Narcissus was the son of the river god Cephissus and the nymph Liriope. Ovid's Metamorphoses tells how he spurned every admirer, including the nymph Echo. Nemesis, goddess of retribution, punished his pride by making him fall in love with his own reflection in a pool. He wasted away, and a flower bearing his name grew where he died. An earlier version has him rejecting a young man, Ameinias, instead of Echo.
That ending makes it look as though the plant took his name. But Greek writers such as Theophrastus and Dioscorides already called a flower narkissos, probably the poet's narcissus. Narcissi were described long before the story appeared. Wikipedia's article on the plant states flatly that there is no evidence the flower was named after the youth.
So where does narkissos come from? The Roman writer Pliny the Elder, who gave Latin the form narcissus, had an answer. He said the plant was named for its heavy fragrance, from the Greek for "I grow numb." That connects it to narkē, "numbness," the root of narcotic. The idea has some chemical logic, since narcissi contain sedating alkaloids and can be poisonous if eaten.
Modern scholars are doubtful. The etymologist Robert Beekes calls the numbness link a folk etymology. He argues that the word's -issos ending clearly marks it as pre-Greek, borrowed from a language spoken in the region before Greek arrived. Its original meaning is lost. The classical narkissos may not even have been today's flower; Etymonline notes it might have been an iris or lily.
English took the plant name in the 1540s. The genus was formally named Narcissus by Carl Linnaeus in 1753, which is why daffodils are narcissi to a botanist. The boy kept his own branch of the family. By 1767 his name stood for a self-admirer, and the psychiatrist Paul Näcke built the term narcissism from it in 1899.