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

Solution 1:

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.

• great minds
  derived from Good wits doe jumpe, which transformed into
  Great minds think alike

• Pot, kettle, black
  Derived from The pot calls the pan burnt-arse, which later became
  The pot calling the kettle black

• devil-may-care
  Originally: The devil may care, but I do not

• second nature
  derived from an ancient proverb: Custom (or usage) is a second nature

• speak of the devil
  derived from Speak of the devil and he doth appear

• the last/final straw
  originally: The straw that broke the camel's back

• the more the merrier
  derived from: The more the merrier; the fewer, the better fare

• when in Rome
  Originally: When in Rome, do as the Romans do

• every man for himself
  Originally: Every man for himself, and the Devil take the hindmost

Solution 2:

I found a noble page with many rhetorical devices described. Included is the paradoxical Inclusion by Omission:

There are two words that may apply.

aposiopesis

An unfinished thought or broken sentence. "If you don't eat up that spinach, I'll..."

ellipsis

Omission of one or more words. "If only pigs could fly!"

Please also see this Wikipedia page on Ellipsis (linguistic).

In linguistics, ellipsis (from the Greek: ἔλλειψις, élleipsis, "omission") or elliptical construction refers to the omission from a clause of one or more words that are nevertheless understood in the context of the remaining elements.