This is something I have been trying to work on lately - just cramming my head full of stuff that can be processed in the ol' background thread.
What I have found is that sometimes, some seemingly innocuous thing I read years ago will pop out at just the right moment, some neural connection forms and all of a sudden I have the basis of a whole rabbit hole of interesting thoughts.
I think it's important to fill your mind with good stuff, things that have the necessary depth and maybe enough subjectivity that they're ripe for that kind of background processing. You are probably not going to have deep interesting thoughts about Twilight (although maybe you will, no judgment.) But philosophy, poetry, great fiction, even religious texts - they're all crammed with symbolism and really reward re-reads and deep thoughts.
A recent example of this for me is Kazuo Ishiguro's book The Remains of the Day. It was mentioned on a podcast I like and so I picked it up on something of a whim during a library visit.
The book is about a butler who served a "great English house" and when I started it I was deeply skeptical that I would enjoy it. But it's a great example of the kind of work that really bears out for thinking on, and since I finished it I've really enjoyed turning things over in my mind. I'm usually a genre fiction guy (sci-fi, horror) and a lot of those books just don't have the same depth.
I don't mean to disparage "light reading" because I'm truly not that kind of. snide intellectual douche. But it's really fascinating sort of watching your brain pick something apart that warrants deep thought, and it's a really cool feeling to be in some discussion with my wife and think "oh my god, that bit from the introduction of book XYZ is so perfect right now, I need to share it and talk about it."
One other book that I've really enjoyed the "post-processing" phase on is The Essential Rumi - the Coleman Barks translation. Those little poems are so rich and full of meaning, and it's seriously rewarding to let your brain go to town on them.