It doesn't matter too much if you are covering it, avoid mdf as that goes funny if it gets wet at all. Ply board is the normal one. Inside a shed you don't need it to stand up to predators just the weight of a bunny.
I would say it depends how much support you are giving the base and how big an area it is. If it's got support bars going across underneath then you can go thinner than if you've got, for example, a 6'x2' piece with no support in the middle. Also depends on your bunnies too - a couple 6kg giants weigh a lot more than two 2kg bunnies.