This page will tell you everything you need to know about how to safely and ethically house a hedgehog.
Cages: Hedgehogs require ample floor space to run around as well as ventilation. Plastic storage tubs are my top recommendation, try to find one with 4 cubic feet of floor space. Attaching multiple storage tubs via 4" pvc pipe can also be done to expand the room. If using a tote, you will need ventilation, holes can be easily drilled into the sides (just make them small enough that hedgies can't escape, less than 1" in diameter) or if your tote is high enough and you don't have any climbable "furniture", then the lid can be kept off. Any metal cage can be used if modified correctly. Hedgehogs suck at climbing, but that doesn't stop them from trying! C&C cages, Ferret/Critter Nation cages, Midwest Guinea Pig Habitat, all can be used with DIY coroplast up the sides. There are many tutorials and Youtube video out there to show you how this can be done.
Multi-levels: As mentioned, hedgehogs suck at climbing, DO NOT use cages with multi-levels, if you need more floor space, get a bigger cage. People try to make ramps safe by adding sides or using flex tubes, usually the inclines are too steep because the cage is too small. Even if you could make it perfect, think about the long term, is your geriatric hedgehog going to be able to still climb the ramp or will you just have to buy another cage in 5 years anyway?
Heating: Open top totes and metal cages can loose heat and make it so your heat source never turns off. Blocking the sides and tops with coroplast, cardboard, or even a cage cover can help keep heat in the cage. Make sure that nothing is ever touching your heat source.
Escapees: If your hedgehog manages to escape its cage or enclosure, don't panic! Start by looking close by and move out from there. Search tight, dark, and/or warm places. If you cannot find your hedgehog, leave food and water out by the cage, they are creatures of habit and have great spatial memory so they will come back to the cage.
Bedding: I use fleece for all my hedgehogs. Appropriate bedding includes paper bedding like Care Fresh or Yesterdays News, compressed pine pellets, compressed paper pellets, and aspen shavings. Cedar shavings should NEVER be used as it can cause respiratory problems. NEVER use puppy pee pads or grass carpets/mats, your hedgehog could tear it easily (burrowing animals like to shred things) and ingest the plastic, sometimes puppy pads have absorbents in them which expand when wet (like baby diapers). Sometimes products like paper shavings or pellets have baking soda in them, do not use these.
Mites: Shavings and paper come with the risk of mites, which is treatable, but preventable. There are 2 prevention methods. Freezing for at least 24 hours (must be a deep freezer, regular freezers usually don't get cold enough) or oven bake about 2 inches deep in a pan at 200 for an hour. Always bring your frozen or baked bedding back to room temp before putting in your enclosure.
Temperature: Hedgehogs require warmth, but not too much, or they try to hibernate/aestivate either is lethal to the African Pygmy Hedgehog. 72-78 F is optimal, some like it on the lower end of that range and some like it on the higher end. You will need a ceramic heat emitter (CHE) bulb and a dome light (at least 8", 10" is better) with a ceramic socket (not plastic). Do not use a red reptile light, blue/black light, or regular light bulb. You will also need a thermostat so that your che will turn on/off as needed. Amazon sells all of this, or you can find them at your local pet store. External heaters can be used if your hedgie is in it's own room or you are comfortable hanging out at those temperatures, most humans are not. I do not recommend heating pads under the cage, this is a recipe for disaster on many levels. NEVER put a heat source directly in with your hedgie, you could cause a burn easily. More information on why temperature is so important can be found here.
Drafts: Where ever you put your cage or enclosure needs to be in a spot that does not get drafts, either from windows or air returns. If you use an external heater, make sure to place it so that it does not blow directly onto the cage.
Summer: Depending on where you live, summer can become quite hot. This means that not only will you need a heat source, but you will also need a way too cool it down. DO NOT let your hedgie's cage get above 90 F. Don't run a fan or air conditioner pointed at or next to your hedgie's cage. There are many thermostats that will allow you to plug in a heat source and a cooling source and it will keep the cage at the temperature you set. If you have central air or a window ac unit, all you will need is the heat source.
Lighting: Hedgehogs are nocturnal but require periods of light and dark. Do not put your hedgehog's cage in direct sunlight. Make sure to provide them with a day/night cycle (typically 12 hrs light and 12 hrs of dark) a light on a timer or leaving your room light on during the day and turning it off at night will suffice. It's normal for babies (and even some adults) to get up during the day to eat or drink. Babies will still sleep most of the night away as well, getting up to explore or run periodically then taking a nap. Do not make your heat source and your light source one in the same, they still need to be warm with the lights off. More information on why light cycles are important can be found here.
Nightlights: Don't use a nightlight, black light, red light, funky party light that changes colors, etc. Your hedgehog will not appreciate your efforts to make them look cool at night. They will be less active without total darkness.
Exercise: Hedgehogs REQUIRE a wheel to run on as a part of their normal, healthy lifestyle. Hedgies are known and recorded to run in excess of 5 miles per night. The only type of wheel I recommend is a bucket wheel, at least 11" in diameter. Carolina Storm Wheels and other bucket wheels can be found on Etsy as well as Facebook groups. Do not use metal wheels, wheels with seams, wheels with sand paper, wheels with covers or carpet, wheels with holes, or saucer wheels, none of these are the right size or safe for hedgehogs. Toes have been lost and feet and spines have been injured using the wrong wheel.
Furniture: Hedgehogs need things to hide in. Cuddle sacks, fleece cubes, igloos, tubes, pvc pipe (at least 4"), other wood or plastic hides can all be used. It's best to get a variety and offer them choices, some hedgies have a preference.
Climbing: Whatever you decide to put in your hedgie's cage, make sure they can't climb onto it to prevent injury and escape attempts. Keeping hides away from the walls can help prevent bar climbing and escapes from open totes.