I'm not sure how this happened and I couldn't get it to do it again yet. But I figured that if someone would be able to figure it out, someone on this forum would.
I was doing a class mission as a Jedi consular after one of the missions when I was instanced in I accepted my reward and without leaving the instance I logged out and onto another toon. When I came back to the consular the mission complete popped up again to accept it again so I did and I got more exp money and the rewards again.
I tried logging out again to see if it would pop up a third time and it did not. I do not remember how long I was long out the first time. But it might have been around two hours, the second time I tried to log out while still in the instanced mission I was only logged out for about an hour. I am not sure but I am going to try this with one of my other toons and see if you can get double rewards after logging out and back in instantly and I'll post my findings. I thought I would put this up here so other people could try as well.