The machine helicopter is a borrowing from “hélicoptère”, first named in French in 1861 by Gustave de Ponton d’Amécourt. His machine did not fly; the first nontrivial helicopter flight would not be completed until 1923, but the name stuck. Hélicoptère was a novel compound of the Ancient Greek words ἕλιξ (hélix, “spiral”) and πτερόν (pterón, “wing”). Some other derivations from those Greek words include, well, “helix”, and “pterodactyl”.

In the intervening time, people reinterpreted the compound as heli- + -copter rather than helico- + -pter. Both rebracketed affixes are productive in English! For example, “helipad” is first attested in 1960, not too long after the first commercially viable helicopter started production in 1942. “Quadcopter” is first attested in 2004 as the media term for what researchers had been calling “quadrotors”. (I delight in linguistic frankensteins, so folks insisting it should be “tetrapter” instead since “quad” is Latin are a fun side note to me.)

More recently, “helicopter parent” became a common phrase in the early 2000s, but the metaphor is a bit older, first seeing print in 1989.