Props is originally hip-hop slang, derived from propers, both similarly meaning due respect.
According to The Rap Dictionary ("the oldest and ultimate resource for looking up hip-hop slang"), props is:
An abbreviation of "propers" or proper respects. A show sits on physical and non-physical props. At an award ceremony the winner gives props: "And I would like to thank...". "Cause this hip-hopper gets props just like a mobster" -- Da Youngstas featuring Pete Rock & C.L. Smooth (Who's the Mic Wrecka)
An older Hip-Hop/Slang Dictionary posted to alt.rap in June 1992 by Lee P Miller defines:
props : 1) (n) Proper respects.
The oldest use I found in alt.rap is from July 11, 1991 by Gregory Vincent Battle:
Now wait a second here. I have to give Hammer minor league props, even
before "Hammer Don't Drown Em". I personally HATED Hammer when I first
heard him. He stole rhymes (worse than Shy D, that fuck), and
inimitable rap ideas ("yeahh boyeee"), and Too Big was more of a black
Pilsbury Dough Boy than a sidekick. Shit, I'd kick him in the side :)
Well, I got free tickets to his first concert tour and I enjoyed it
completely. His show was pure energy. I think he might overdo it now
with his 60+ dancers/backup rappers/DJ's. But hey, I still don't like
him enough to buy one of those wick-wack rekkids.
The OED also says it's a shorted plural form of proper with a first quotation from the Chicago Tribune (July 1990):
I was one of the first female rappers, but I've always gotten my props.
The OED says propers is similar African-American slang for due respect, first quoted in the Chicago Daily Defender (January 1971):
A level of existence which affords each black man his propers—dignity, pride,..and the ability to govern his destiny.