Balance Totem
I've used my time just studying and designing a way to make a the base of the desk toy achieve that balancing motion that we want. First basic design plays plays with a design that puts all the mass of the spindle at the bottom half as the top curves up.
A center of mass is shown, however it is lower in reality due to not having a uniform mass 
Different infill densities were used to lower the center of mass closer to the bottom.
40%
40%
6%
6%
Printing and Results of Prototype 

It is very stable, however due to 3D printer's use of supports it makes the bottom too rough and jagged to actually return to the straight up position. Using sand paper on the bottom will fix this to via making it smoother. 

Lowering the center of gravity would be very beneficial since the range of stable positions could be larger.
Could do this via increasing lower density, which will result in longer prints for higher densities. 
Another method is designing a cavity for a weight to be placed mid print, which is simple but has the cost of buying weights. 
 
Next:
Not the most artistically gift but I can work on the top "totem" part of the toy by making it more than a simple rectangle. Adobe Illustrator/ Blender have 3D model features that are intended for artistic looks that engineering focused ones don't have. I have limited experience with them but I can have an attempt if that's the direction you'd like me to go in. 

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