Greeks named them for their purple dye ('phoinix') — the Phoenicians called themselves Canaanites.
An ancient Canaanite Semitic language spoken by the Phoenician civilization of the coastal Levant (modern Lebanon), famous for creating the alphabet that is the ancestor of nearly all modern alphabets; also used for the people and their culture.
From Latin 'Phoenīcius,' from Greek 'Phoiníkē' (Φοινίκη, Phoenicia) and 'Phoínikes' (Φοίνικες, the Phoenicians). The Greek name almost certainly derives from 'phoînix' (φοῖνιξ), meaning 'purple-red' or 'crimson,' a reference to the famous Tyrian purple dye extracted from the murex sea snail — the most valuable commodity of the Phoenician trading empire. The word 'phoînix' also meant 'date palm' and was the name of the mythical phoenix bird, all connected through the color purple-red. The Phoenicians called themselves 'Kenaʿani' (Canaanites), not Phoenicians — the name is entirely Greek.
The Phoenicians never called themselves 'Phoenicians' — that is a Greek label meaning 'the purple people,' after their lucrative murex dye trade. They called themselves Kenaʿani (Canaanites). And while their trading empire has long vanished, their greatest invention — the alphabet — survives in almost every writing system on Earth. The letters you are reading right now descend, through Greek and Latin, from the Phoenician script developed in Byblos around 1050 BCE.