How to Bring a Scared Cat Out of Hiding

I was recently reminded of how much patience this requires while on a Kelly's Pet Sitting visit. The female cat in question ("Hen") was ill and didn't have a cat or dog companion, so it was of the utmost importance that I get her out from under the bed and make friends with her. 
 
In my experience, there's always something that will catch a cat's attention and bring them toward you. Usually this response is instilled over time by the pet's parents.