Before the Mission
Before the mission, depending on your class of civilian your role varies. For example, if you are a botanist, you can say that your character has made an important discovery on plant life that could improve relations on another planet. Or, if your character is a cook, you could say that she was cooking a new recipe that went horribly wrong and ended up poisoning the entire crew that had eaten in the cafeteria. Mainly your job is to have the ship run as smoothly (or not as smoothly) as possible.
During the Mission
During missions it is a little harder to get your character into the mission. However, there are quite a few creative ways to get involved. First is perhaps allowing your character to be captured during an alien invasion. Secondly, have your character hear or see something that will drive the mission forward. If all else fails, create a PNPC that can work in a department.
After the Misison
Once everything has been finished on the mission, back to square one right? Wrong, there is still so much for your civilian character to do! When the ship was attacked, did one of your close friends get injured or die? Sim the emotions of the character. It will be a good read for everyone as well as lead to further character development. Also as a civilian, you get to tell about what happened behind the military lines. What was seen? How did it affect your character and the outcome of the mission? All of these rules should help you get started on simming this duty post.