First attempt: elastomer that bounce back. The best is natural rubber. Some artificial rubbers do bounce, others absorb. Foams help where rubbers are too hard.
What I dislike: if you fall from 0.5m and rebound over 0.05m you get >10g, rather 20g or more, fed by your knees, so you're hurt.
Second attempt: feed some deformable envelope with a pressured gas, for instance air, like many bouncing balls are made, and also tyres: tennis, soccer, basket balls. More elastic, lighter, cheaper. You can build your thingy thicker for a less hard shock. But don't expect miracles.
A deformable envelope often means fibres in an elastomer, possibly with two envelopes in an other like many tyres have, and a valve for inflating. Consider existing hardware like bicycle inner tubes, fit it in a shell.