What’s the least I could get away with that would run Sonic Pi and output (any) audio?
Sonic-Pi was designed partly with the Raspberry in mind. Maybe a simple vanila Rasp could do the work. Cheap keyboard, cheap mouse, old monitor and you’ll be running it in no time.
EDIT : if you already have other equipment, it may be possible to run it headless. I don’t know anything about that but I’m sure that people tried it already.
You can use VNC to do a remote desktop to the Pi if you want to run it headless. I have a handful of them on my desk and often either ssh in (for command line) or VNC in (for desktop) instead of switching them to a monitor. If you go with a KVM or HDMI switch approach, there is a setting in the boot.ini that forces it to assume an HDMI monitor even if it doesn’t detect it properly with the KVM’s monitor emulation.