What can cause left ring finger pain that turns a little blue?

Raynauds. This is a condition in which the arteries in the finger clamp down and the finger temporarily looses its blood supply. Avoidance of cold is the best treatment.
Any disturbance in . Circulation, nerve input, temperatire or even pain and sweliing can cause vasoconstriction when arteries get too tight or vasocongestion when the outflow through small veins is disturbed. Both of these can cause a color change in the affected finger. Even simple things suchas a tight ring or a trigger digit can affect the color and turgor ( feeling of fullness) in a finger.