Is it supposed to be Just Cause, as in blowing up that building was a just cause.
 
Or is it Just 'Cause, as in I blew up that building just cause I can.

It's about a revolution ergo first answer is correct. Second is just a bad pun everyone likes to use. 

I'm guessing both.

" I'm guessing both. "

Characters use the former definition of the phrase in the dialogue ("I think that's a just cause," or something like that) but in all likelihood it's intended to be both.

The first one. The second one is the result of americans trying speak

" It's about a revolution ergo first answer is correct. Second is just a bad pun everyone likes to use.  "

Just cause somebody thought it was a just cause to blow up a building doesn't make it a just cause.

just =   used especially of what is legally or ethically right or proper or fitting;
cause = campaign: a series of actions advancing a principle or tending toward a particular end;
 
i think these are the definitions the developers were striving for when naming the game.

Why did I blow up the building?  Just Cause.  But that isn't really a Just Cause.

" The first one. The second one is the result of americans trying speak "

both

Just Cause as in 'just purpose', the 'just because' interpretation seems downright idiotic.