If I want to create a bunch of Tor relays running Linux with various VPS providers and want to save on cost, what is the minimum hardware config that I can get away with while still effectively relaying traffic?
A lot of us are running on Raspberry Pi. Almost every hardware setup can handle some amount of traffic. One just dials down the amount of bandwidth one is willing to advertise or accept in the configuration file.
There are a whole range of relay capabilities. You can see a list of their bandwidths at http://torstatus.blutmagie.de/. The fastest relays obviously have fiber connections and enough RAM and CPU to handle the traffic.
You can monitor your node to see if it is getting crushed by the amount of traffic it is receiving. My Raspberry Pi is able to qualify as a fast exit node, handling about 200KB/s. In this case, the machine is CPU constrained. For other setups, they are bandwidth constrained.
The Pi has 512 MB RAM and the processor is by ARM. The whole machine draws less than 5 Watts, about what a nightlight draws. And it is very stable.