meaning - Is there a word for using half an idiom or saying?

Sometimes, when engaged in conversation, English-speakers will say the first half of a well-known saying or idiom, trailing off at the end or punctuating it with a shrug. For instance, they might say "Birds of a feather..." or "When in Rome..." and leaving the "flock together" and "do as the Romans do" completely unsaid, when arguably that unspoken portion was the whole point of their statement in the first place.

Is there a name for such abbreviated usage?

Answer

The technical term, anapodoton is an appropriate, and perhaps more accurate alternative to ellipsis

An anapodoton (from the Greek anapodosis: "without a main clause") is a rhetorical device related to the anacoluthon. It is a figure of speech or discourse that is an incomplete sentence, consisting of a subject or complement without the requisite object. The stand-alone subordinate clause suggests or implies a subject (a main clause), but this is not actually provided.

As an intentional rhetorical device, it is generally used for set phrases, where the full form is understood, and would thus be tedious to spell out, as in “When in Rome [do as the Romans].” or “If the mountain won't come to Muhammad [Muhammed will go to the mountain].”

You might also call them clipped, or simply, shortened idioms. Here is a list of shortened idioms that I found scouring the ELU archives.