I have a big vein in my lower left abdomen. Why only one side?

Seek help. The isolated vein may be collateral venous drainage, especially if you have had previous blood clot in the left leg, involving the major portion called the iliac vein. The blockage of blood return through the iliac vein resulted in the body opening other channels for blood return. If you have not had a prior blood clot, it may not be significant. Seek help from your doctor.