What is the cheapest hardware device that can run a proper bitcoin client that can participate in the P2P network?
You can use any computer that fulfils the minimal requirements:
- Most important is sufficient storage for the blockchain (currently ~9.5GB), you probably want a multiple of that for some peace of mind.
- The computing power necessary to just run a (non-mining) client is insignificant, any computing device available on the market today should do. If it isn't particularly powerful the initial sync might take longer, but after that the effect is neglible.
- Internet connection with sufficient bandwith to keep up with the network.
An old computer or Raspberry Pi would be sufficient, whereas the Raspberry Pi will probably have a much better energy efficiency and thus should end up being cheaper in the long run.
A comment above suggested that a virtual private server might be even cheaper, however, if you want to store any significant amounts of money on your bitcoin client, you might be more comfortable to do so on hardware that you have complete control over.