What is the word for two-part phrases where the second (or first) half drops off?

This phenomenon is generally called an "ellipsis". But this word can also be used to describe a typographical sequence of three dots ("..."), often used to indicate missing speech rather than describe an incomplete saying.

If you want to be more precise (and technical) and avoid confusion, you can use the word "anapodoton".

UPDATE

Thought you might like some more examples: http://en.wiktionary.org/wiki/Appendix:English_examples_of_anapodoton

"Abbreviate" DOES mean to shorten in part, but I think what you are trying to find, specifically, is the word for a historic trend to abbreviate certain phrases, which has a history dating as far back as ancient Roman scripts. http://en.wikipedia.org/wiki/Abbreviation#History

Ironically, because it is so prolific in our language, there doesn't seem to be any word at all to describe this historic tendency. The best I can come up with would be just calling it "abbreviation" or, as suggested above, "phrase abbreviation".

If, however, you mean the 'forgetting' of the second part (or omitted part) of the phrase, while the meaning remains...I'm afraid I can't help you there. But you could perhaps emphasize that as what you're trying to find.