I have an African hedgehog and he uses a wheel every night and I would like to harness his energy in some way.
Is it possible to create a system that would allow me to transform and store it? If so how?

For a 70 kg human, it would be realistic to get 100 watts out continuously (over a timescale of hours).
For an African hedgehog, that has a mass of 450 grams (according to Google), if they have same power-to-weight ratio as humans, you would get less than a watt out.
That could perhaps run a very dim flashlight as long as the hedgehog is running. However, these days flashlights are typically rather bright, many watts, so even a bright flashlight won't have enough power.
A smartphone LED light has around 50 lumens and requires about the same power the African hedgehog would produce. If you are in a dark room, and turn on the smartphone LED, you will probably see the room is still very dark.
Also, in bicycles there are hub dynamos for powering headlights from pedaling power and they generally have only about 50% efficiency. They are usually 3 watt units (6 watts in, 3 watts out). So assuming your dynamo would have the same efficiency as bicycle hub dynamos, the electric power your African hedgehog plus poor efficiency hub dynamo would give would be maybe third of a watt. That wouldn't be plentiful enough even for a smartphone LED.