How normal is it for a varicose vein to hurt after giving birth?

Yes: During pregnancy your estrogen/progesterone levels are elevated. Afterward, while lactating, your prolactin levels are up. These hormones can allow varicose veins to stretch larger and be more engorged than at other times, which also stretches nerves in vein walls, which hurts! wearing compression stockings helps, and when done breastfeeding, see a phlebologist about closing these abnormal veins.

Not unusual: Pregnancy and childbirth put a tremendous amount of stress on the venous system of the legs. This is due to the increased pressure in the pelvis which limits the vein drainage from the legs. I would recommend an evaluation by a vein specialist

Varicose vein pain: There is not a "normal" when it comes to "varicose veins". Pain within your varicose veins after pregnancy could indicate something serious such as a blood clot or phlebitis, so it is always wise to see a vein specialist (ie vascular surgeon) to examine your legs when symptoms develop. It may be that the abnormal veins just got larger as a result of pregnancy, leading to more symptoms.