Upon recently thinking about the Force Awakens, a few custom skills popped into my head. These were inspired by very specific events in the movie, and their level of utility might vary, depending on the situations and rules your group generally encounters.
Custom Skill: Luck (Willpower)
Some people are just born lucky, but the more they push their boundaries, the more their luck seems to hold. This isn't the all encompassing will of the Force so much as being able to do the near-impossible in situations where others might fold.
Use of the Luck skill is extremely limited. It is only utilized when a character is attempting an Impossible Task (as detailed on page 18 of the Edge of the Empire Core Rulebook). Essentially, the skill reflects the ability of the user to ignore the probability of success and attempt almost anything in a pinch.
The Luck skill is one that I wouldn't assign as any specific careers, as it may just be part of the mystique of some characters, regardless of their backgrounds.
Custom Skill: Sanitation Systems (Intellect)
Nearly any relatively advanced settlement with biological inhabitants is going to need sanitation systems of some sort to function properly. Some individuals learn the routine chores related to keeping those systems functioning, and may learn other useful, related details as well.
A Sanitation Systems check may allow a character skilled in its use to learn where various access ports are located, what kind of biological hazards might be present in the system and what kind of countermeasures to take, and how to perform basic repairs and upkeep. They may also make checks to perform acts of sabotage on those systems.
The Sanitation Systems skill is one that would likely be a good career skill for Colonists and Technicians in Edge of the Empire,and Engineers and Soldiers in Age of Rebellion