You'd have a hard time keeping a wild animal as a proper pet for the sole reason of it being wild! If you're lucky enough to have a wild hedgehog living near you, you could put specific food out for it. Or if your garden is secure you may be able to 'rescue' a hedgehog that's been taken out of its original environment for safety reasons. To do this you need to contact your local wildlife rescue centre and enquire.
If you're just looking for a pet which lives outside, how about a tortoise? Or a rabbit? With either of these I'd definitely recommend reading up about the animal first to see whether it suits you and the environment you live in.