In medium saucepan combine all ingredients except eggs. Simmer over low heat, uncovered, until onion is tender, about 10 minutes.
Arrange eggs in each of two 1-quart jars with tight-fitting lids. Pour 1 cup vinegar mixture over eggs in each jar. Cover and refrigerate several hours or overnight to blend flavors. Eggs may be stored in refrigerator up to 2 weeks.
MAYONNAISE
About 1¼ cups
2 egg yolks or 1 whole egg
2 T. vinegar or lemon juice, divided
1 tsp. sugar
1 tsp. dry mustard
½ tsp. salt