What might be the cause of pain in a small non swollen vein in right hand between thumb and first finger?

Likely muscle/joint: If there is no swelling of the vein or associated redness, this is unlikely to be a vascular source of discomfort. The area between your thumb & first finger can become sore from prolonged movements like grasping, pulling or holding something, esp if repeatedly. Often discomfort doesn't peak for 24-48 hours after such activity. Try to limit such movements to see if improves.