"Seriously, the idea that someone would need advice on how to 'start working on a side project' seems almost bizarre"Well it isn't bizarre in context right? There are lots of people who have spent the first 18 - 20 years of their life what to spend their time on. A significant fraction of them have never even begun to work on things that weren't "part of the plan" as it were. So coming up with a side project is a daunting task.
The best advice is simply do something you haven't done before, code a compiler, code an interpreter, code up a javascript library, design a web site, take pictures of every bird in your back yard, dig up some dirt and sequence the DNA of things you find there, plant a garden, take an art class at the local community center, volunteer to sort food donations at a food bank, pretty much anything you haven't done before try it.
That way you can discover things you are really interested in, and things you really aren't interested in. Since theses experiments are things you never have to do again if you don't like them, go in with the attitude of "Hey I can tough it through anything once, because it's only once."
Its pretty simple advice and I am always amazed when someone hasn't thought of it already, but I have met a number of people who have graduated from college and then paniced with no idea what to do because their 'plan' ran out.