Editing voice clips of someone famous into a full song (original or cover). The more elaborate version of Stupid Statement Dance Mix, which takes only a few statements and loops them.
The arguable forerunner to the modern Voice Clip Song is The Firm's 1987 hit and massive Ear WormStar Trekkin'. It didn't actually use clips from Star Trek itself, instead having the band members sing each character's trademark phrase (even if some of them were, ironically, examples of Beam Me Up, Scotty!) along with the music.
The most epic and heartwarming example of this is "Kidung Abadi" (Eternal Ballad); it was created by Erwin Gutawa for a concert honoring the 5th anniversary of the death of legendary Indonesian singer Chrisye (Gutawa had collaborated with him before, but never got to write a song for him). Erwin had his sister Gita write the lyrics, and then worked with a team to splice them together using vocals from Chrisye's master tracks. For the live performance, Jay Subiyakto also spliced together concert footage of the singer himself (to make him "sing" the new song) as a finishing touch. The result was three months definitely well spent.
Remixer Rx has made a name for himself doing these with politicians:
"Deify" by Disturbed starts with clips of one pro-Bush source, one comment on a repressive government, then clips of Bush himself from his 9/11 speech.
A Glorious Dawn, mixes and sets a tune to parts of Carl Sagan's Cosmos and Stephen Hawking's Universe. Notably, entire phrases from their respective shows are used. It's rather awesome. And thanks to Jack White's record label Third Man, it was actually given a limited release on vinyl, with a copy of the diagram from the voyager golden record etched on the b-side.
Most of Negativland's out out falls under this category.
If you've ever played Kingdom of Loathing and wondered why using or buying eleven of an item gets you the message "That's ridiculous. It's not even funny", it's from the Negativland song "Time Zones". In turn, that was from a talk radio show clip discussing the Soviet Union (which apparently had eleven time zones).
I Don't Like It is a mix of various statements uttered by controversial Australian politician Pauline Hanson (the video features performer/mix artist "Pauline Pantsdown" lip-synching the performance)
Finnish rap artist A-Tyyppi has mixed phrases from sports commentator Antero Mertaranta into a truly hysterical dance song. It's still regularly played whenever Finland manages to kick ass at ice hockey.
One early example would be Paul Hardcastle's 1985 single "19": There are some professional backing vocalists singing a few lines, but the vocals are mainly clips from a documentary about The Vietnam War.