There seems to be quite some possibility to acquire fpgas on ebay.
Most of them are from damaged boards where you would have to reball them yourself, but I think we woudn't necessarily need that.
My thought is get them off the board with a hot air gun and glue several of them on a big heatsink upside down with a heat-conducting glue.
They possibly couldn't be used for anything else after that but what the hell, they are most likely living their rest of their lifespan generating bitcoins anyway...
Then solder wires directly onto the pad were the reballing would take place, that could be quite difficult but I think it would be possible.
The optimal way would be to devise a system were a minimal amount of wiring is required, only jtag + serdes nodes to get the data though. (or possibly even with the jtag backchannel, would be even more awesome)
The only real isse I see here is supplying power, since they sometimes require different levels and designing differnt power converter circuity everytime would not be very efficient and would require too much work.
Possible sources:
http://www.ebay.com/itm/Xilinx-Virtex-5-XC5VLX110T-64-x-Samsung-XCK9F8G08U0M-PCB0-8GB-Chip-Recovery-/260871631798?pt=LH_DefaultDomain_0&hash=item3cbd2933b6(might be a good board in case anyone wants to grab it and run with it go ahead)
http://www.ebay.com/itm/Lot-100-pcs-XC2V250-5FG456I-Xilinx-PCB-BGA-FPGA-/270846090663?pt=LH_DefaultDomain_0&hash=item3f0faf5da7100 small small old virtexes, might work without unrolling.
What we also would need is a design which can be quickly adopted to many different chips with different I/O configurations.
I think if this works out this could very well be the next big thing in bitcoin mining...