Phones don't have to be made for prepaid service, but they do have to work with the provider in question.
Most providers sell new phones pretty cheaply, but it's definitely possible to get a used phone.
You need first to decide what provider you want to use - then it will be a lot easier to decide what phone to get. Phone carriers have different technologies that are not inter-compatible. Also, many phones are locked to only work with the provider for which they were sold, so even if the technology is the same as the provider you want to use, you may not be permitted to use the device because of the lock.