Hedgehogs in the wild usually cover a lot of ground in their search for food. Pet hedgehogs need lots of room to move about too. Two square feet (e.g. 1 foot by 2 feet) is sometime quoted as the minimum floor space for a hedgehog, but this should be considered an absolute bare minimum, and only used if you have a wheel and give your hedgehog ample time to roam around outside the cage for exercise. It would be much better to aim for a minimum of about four square feet (e.g. 2 feet by 2 feet).
A hedgehog cage needs to have a solid floor, so avoid any cages with wire or wire mesh flooring (hedgehogs may catch and injure their legs or feet on wire floors). Cages should not have any sharp edges or spaces in which a hedgehog could get his or her head stuck. The cage must also be secure to prevent escapes.
Good ventilation is necessary to keep humidity levels down and to prevent ammonia (from urine) and odor from building up in the cage. Wire cages offer the best ventilation.
Ease of Cleaning
This one is fairly self-explanaorty, but do not underestimate its importance. Your hedgehog's cage will need frequent cleaning, and a large, heavy or awkward cage will make this chore very unpleasant.
Many owners use wire cages since they are quite readily available and these have the the advantage of good ventilation. In addition, they are usually pretty lightweight and easy to clean. However, few are made specifically for hedgehogs so you need to be very particular when choosing a cage. Avoid any cages with wire flooring (or if absolutely necessary, cover the wire with wood, plastic, or a Vellux blanket cut to fit securely). Additionally, cages large enough for hedgehogs might have wire spacing that is too large for safely housing hedgehogs (look for ferret or rabbit cages with spacing of 1 inch or less). There are a couple of manufacturers producing wire cages particularly for hedgehogs: Martin's Cages (choose from the larger ones - the Hedgehog Home is very small) and Hedgehogs by Vicki. Some people recommend multilevel ferret cages, but the height of these cages and the platforms make me nervous about falls (from the platforms or from climbing the sides of the cages).
Aquariums are okay, but you need a large aquarium (i.e. 30 gallon is a good minimum) and a wire mesh top. The major disadvantages are the lack of ventilation, and aquariums are heavy and awkward to clean.
Many owners have gotten quite creative with creating cages out of plastic storage bins. With some slight modifications, you can make a large cage for little money out of clear plastic storage containers (solid color containers would be quite dark for a hedgehog). The biggest downside is again ventilation. Holes can be made in the sides and lid with a soldering iron or drill, but it is difficult to provide enough holes to provide great ventilation. If you get a deep enough container and don't have anything (including water bottles) around the walls that the hedgehog can use for climbing, you may be able to get away with having no lid. Alternatively, you can fashion a lid out of hardware wire mesh or screen material, either on its own, or attached to a large opening cut in the lid of the storage container. The Michigan Hedgehog Owners Group site has an ingenious idea for a two-container home with instructions. Hedgehog Valley suggests cutting panels in the sides of the container and fixing wire mesh or plastic canvas over the openings to aid ventilation in this type of home. With creativity, these homes can work quite well.
I have seen other ideas for cages, such as wading pools (solid plastic with high sides) and home-made wooden cages. As long as a cage is large enough, escape proof and safe, ventilated and easy to clean, then your imagination is the limit.