The Wikipedia: BeagleBone Black is $18+ Linux computer -- 1 GHz ARM Cortex-A8, 512 MB DRAM, Ethernet, etc. Sone versions, alibaba.com, are sold without HDMI, USB, less bulk But these vendors don't supply design files, as they are required
Replicape and BeagleBone
The Replicape was apparently originally designed for the thing-printer (formerly HipsterBot), but it should work with most RepRap hardware.
Cape-RAMPS For BeagleBone: CRAMPS
The design is intended to provide a simple low-cost board to interface the BeagleBone to typical desktop 3D printers, mini-mills, and other CNC machines. The design is inspired by the simplicity and success of the RAMPS board for the Arduino Mega, and borrows from the RAMPS-FD design for the Arduino Due.
TheBeBoPr  Some firmware is not open-source. No design files available.
Strange circle-shape PCB. T-Bone
Other 3D printer capes for BeagleBone
A two layer through hole [RAMPS] derivative would be easily designable.
Apparently several other people have built 3D printer capes or other CNC capes for the BeagleBone. "An unofficial list of various 3D Printer, CNC, and motion control related capes".
Other 32-bit 3D printer controllers
Vaporware Electronics lists many other 32-bit processors and boards designed around them that people are considering for 3D printer control.