I personally recommend a small rabbit or guinea pig cage with a plastic bottom and wire sides and top or an extra large plastic tub that has been modified to accommodate a hedgehog. I am currently selling modified tubs that I personally use to house my hedgehogs.
I am happy with this type of housing, because it provides great ventilation and is lightweight. If you are using a plastic storage bin type of tub, you will need to cut holes for ventilation. These bins are very easy to keep clean. The nice thing about the bins is that if you are handy, you can build passageways and tunnels from one bin to another and create levels and all kinds of interesting hideouts. As I’m writing this, I’m thinking of hedgehog playroom ideas! That might be fun. If you have the space, you might also consider buying a solid plastic wading pool (for little kids), not the flimsy ones. You can put all kinds of toys in it for your hedgehog and it would make a good play area. Aquariums should be avoided since they do not provide enough ventilation and are more difficult to clean.
Your hedgehog will also need a hiding spot. You can use Pigloos (Pet Igloos) made of plastic with an opening for them to crawl into and sleep. They also sell covers for the Igloos to provide more comfort and warmth, although these are not necessary. Other things to try would be shoe boxes and tissue boxes.
Hedgehogs need to stay at room temperature, between 75-80 degrees is recommended. If the temperature drops, they may try to hibernate and prolonged exposure to cold could prove deadly. If the hedgehog will be in cooler location, you will want to purchase a small animal heating pad. It can be placed under the cage.
Hedgehogs are solitary creatures and I do not recommend housing two together, although it can be done. Hedgehogs can easily injure each other; Eye injuries are common. Never keep a female and male together unless you intend to breed them.