After printing and using 3Dworkshop's vibration dampers (https://www.thingiverse.com/thing:2557700) for a while, I realised that the PLA I used was so stiff that it wasn't really doing anything much. Hence I decided to design my own vibration damper that is meant to be printed in flexible PLA.
I have included two versions with different thickness (to provide different level of stiffness/damping depending on how flexible is your PLA). The one depicted in the pics was the 1.1mm version. It is printed using Hobby King's flexible PLA (which is still kind of stiff, compared to, say, Ninjaflex).
Other flexible materials such as TPU might work, though I have never tried them.
You may need to play around with the extrusion width (I used 0.52 in Slic3r) to control the number of walls for each of the layers. Note that the stiffness is probably also affected by your choice of extrusion width (which determines whether there would be gaps between the inner and outer walls). I deliberately left some gaps between the walls in some of the layers to ensure that there is still some bounce in the final print. In Cura, you can force this behaviour by turning off filling of the gaps.
I designed the basic shape in Onshape and played around with the thickness and printed a few samples until I get the bounce and flexibility that I'm happy with.
The video shows the amount of play left in my current print (the 1.1mm version) .