Un coup de fil

It's 1975, and Jeannette has just settled into a steaming bubble bath when the telephone rings. She leaps out, slips on the bath mat, races dripping through the apartment, tangles herself in the impossibly long curly phone cord, and finally gasps, "Allô ?" Only to hear her friend cheerfully announce, "Je voulais juste te passer un petit coup de fil pour te dire que j'ai retrouvé mes clés !" -  "I just wanted to give you a quick call to tell you I found my keys!"  The "quick little call" lasted fifteen seconds, but it left a trail of wet footprints across the apartment and a very confused cat wondering what on earth had just happened.  

That's the spirit of un coup de fil: not a long conversation, just a brief, spur-of-the-moment call that somehow requires maximum effort from the person answering.
In those days, to reach your friend across town, you didn't send a text, leave a voicemail, or fire off a dozen emojis. You sent a little burst of electricity racing down a copper wire. If you were not planning a long conversation, you were just giving the wire a quick tap. That quick "tap on the wire" became un coup de fil, and a few decades later, the expression is still alive - even though the fil (wire) has all but disappeared. But why a coup (a blow, strike, or hit), you might ask?

Origin

Of course, no one was actually whacking telephone wires with a stick. Here, coup is used in one of its many figurative senses: a quick, brief action. French uses coup this way all the time:

• un coup d'œil - a quick glance
• un coup de main - a helping hand (literally, "a hand strike")
• un coup de téléphone - a telephone call
• un coup de frein - a tap on the brakes
• un coup de balai - a quick sweep with a broom

So un coup de fil was simply a quick use of the telephone wire.  A brief call.

What's especially amusing is that we've kept saying un coup de fil long after telephone wires disappeared from our daily lives. Today, your call might travel through fiber-optic cables, cell towers, satellites, or entirely over the internet. Yet French speakers still say Je te passe un coup de fil, even if neither person has seen a telephone wire in decades. It's a bit like English speakers saying "hang up," even though phones no longer hang on hooks.

Examples

N'oublie pas de me passer un coup de fil la prochaine fois que tu vas en ville; on essaiera de se voir.
Don't forget to give me a call next time you're in town; we'll try and get together.

Elle passe toujours un coup de fil à sa mère quand elle voyage, juste pour l'assurer que tout va bien.
She always call her mother when she's out of town, just to tell her she's fine.