The thinket makes a very little bit of noise, much quieter than a pen clicking; you can prevent it by moving it more slowly. Spinners make an almost inaudible whirring. The weights of Begleri can tap against each other and make noise, depending on how you use them. A spinning-coin/coin top is quieter than most any other top when it lands. These wooden "fidget puzzles" https://www.amazon.com/Toysmith-Fidget-Puzzle-Discontinued-manufacturer/dp/B00YRZ5QTE/ref=sr_1_19?ie=UTF8&qid=1476022572&sr=8-19&keywords=fidget are basically totally silent unless you try to make noise with them, and can easily be concealed in the hands so it just looks like empty-hands fidgeting.
Particularly for people who struggle with anxiety issues, i have one of these and its a lifesaver when im in a waiting room or something like that. It seems like a child's toy but honestly it's amazing. Keeps your hands busy and somehow allows me to focus a bit more. I'm in the UK, and they sell these (and things like them) for £1, £1.50 in Hawkins Bazzar, not sure if that exists in the US, but worth metioning.
http://www.amazon.com/Toysmith-TSM6496-Wood-Fidget-Puzzle/dp/B00YRZ5QTE
(Someone correct me if I'm wrong - I don't have a ton of experience with hinges.)
Usually, two boards hinged together will only be touching in the closed position, and one will swing away from the other when opened (pictures: closed - no gap, open - gap). You want the ends of your boards to be flush in the open position, and one end flush with one side in the closed position. This is a problem because the corners will prevent one board from rotating around the other. Normally it would be solved by chamfering the edges (for your case, swap "closed" and "open" and ignore the lid), but that would leave you with a long groove when your box is open flat. I think you would have this problem with any traditional hinge.
I can think of two other solutions, one fun and one practical:
Have you ever had one of these toys where the blocks are joined with an elastic band, so you can pull them apart, rotate one piece to a different side of another, and they snap back together? That could be cool!
Can you make the base (the middle piece and the pieces you want to hinge to it) from thicker plywood? Then you can use barrel hinges or invisible hinges to connect them. Check out this photo, except "closed" is your open, "partially open" is your closed, and ignore "fully open."
(You may also want to use thicker plywood for the 900x136 ends if you plan to put latches or a handle on them.)