Remix others’ ideas
Take a chord progression you love and write a new melody to it. In jazz, this is called a contrafact. Chord progressions are not protected by copyright, however, be careful not to steal the chord progression as well as other elements like groove, instrumentation, tempo, etc.1 Jeff Tweedy says if he can still recall the original song, he discards or alters more.2
Use samples (but make sure to clear the licensing).
Write new lyrics to an existing melody.2 Or write a new melody for existing lyrics.
Keep a “morgue file” of things you’d like to remix later.3
Lastly, if you’re going to totally rip something, you’ve got to add something to it to try to make it better.4 Shakespeare ripped things, but he made them better using the figures of rhetoric.5 He ripped sections of Thomas North’s translation of Plutarch’s Lives of the Noble Greeks and Romans to be used in The Tragedy of Anthony and Cleopatra, and simply added Alliteration.6