They... Them... The powers that be are giving away every movie prop ever used. You are in the unique position of being first in line. This means that you get to pick any prop you would like.
Which would it be? As a side question, what would you do with it?
A couple of caveats...
1) Everything is a prop. It doesn't necessarily work or is high quality. A light saber is not going to cut anything. The bearer bonds from Die Hard are just pieces of paper, etc. 2) Sets and sound stages are not being given away. 3) You are required to sign a waiver promising you will not turn around and sell the prop. (You can give it away, otherwise you have to keep it)
If television shows are eligible, then there's no question: A hero type II phaser used in the first 8 episodes of the original Star Trek (i.e., one of the black and white ones -- not the later gray or blue ones).
Limiting to just movies, if miniatures qualify as props, then the original large-scale Millennium Falcon from Star Wars. If miniatures don't qualify, then I would be happy with a nice stormtrooper blaster (a modified Sterling, complete with the Hengstler counter and Azimuth scope), or maybe the hero version of Deckard's blaster from Blade Runner.
(Note: In prop jargon, "hero" refers to a nice, detailed model that sometimes has moving/functional parts. These are used for close-ups. Non-hero versions are used for most other scenes, and rubber castings are often used as stunt doubles.)
"We're kind of on the level of crossword puzzle writers... And no one ever goes to them and gives them an award." ~Joe Strummer sscce.org