Currently I have a working prototype costing around $15:

The temperature sensors are DS18B20, they fit snugly located in custom 3D-printed ring and wristband. The 3D models are available here: ring and wristband add-on. They were created with FreeCAD and here are the project files in case you want to modify them: ring, wristband.

The board is a C.H.I.P., an insanely cheap ARM-based single-board computer. In the current version, the board simply records temperatures and broadcasts them to a Wifi connected laptop. There is no detection of Raynaud's crisis for now! I need to gather more data to identify reliable signs of an incoming crisis.

The cable that you see leaving the picture from the right edge is the USB power supply to the board. In the next iterations, a battery will be embedded. I am also thinking about dropping the C.H.I.P. altogether and switch to an ESP8266, because that would use less space and less power.