1.To effectively dispatch anyone ignorant enough to attempt an assault upon you, or a contact of high importance.
2. To make someone look breathtakingly inept during a physical confrontation.
3. To "rock the shit" as Brian Fantana put it.
"A bunch of wannabe gangsters tried to jump me, and I jason bourne'd them. Needless to say it was spectacular."
To claim to have no memory of something by placing the blame on a government organization, rather than one's own forgetfulness.
Spouse: "Did you not remember that today is our anniversary?"
You: "Sorry, significant other, but I Jason Bourne'd all about it"
Spouse: "Did you forget the milk, again?"
You: "Whoops. My bad, love of my life, but I Jason Bourne'd as I walked past the dairy aisle."
Destroying the Evidence:
Spouse: "Did you get rid of the body like I told you to?"
You: "Damn. I'll admit, ball and chain, that I'm a highly-trained assassin that works for a shadowy government organization that I can recall almost nothing about... i.e. I Jason Bourne'd the corpse."