What is the word for adapting a saying to another purpose

One obvious candidate word would be repurpose. From NOAD:

repurpose (v.) adapt for use in a different purpose

One blogger used this word in her piece about the term Black Friday:

...retailers have since repurposed the phrase and given it the more positive connotation that it has today.

Here's another example:

That’s why Liu says she helped coin the term, the ‘China Dream.’ Even though the Chinese government has repurposed the phrase to talk about rejuvenating China, initially it was supposed to be a response to the American Dream and the cornerstone of Liu’s project.

What you describe sounds very much like metalepsis.

From the wiki: "... a figure of speech in which a word or a phrase from figurative speech is used in a new context."

The example they give is "I've got to catch the worm tomorrow", which is adapted from "the early bird catches the worm", but changes the context and arguably the meaning to some extent.