Here are three suggestions, all slightly different.
This is the most classic sauce for oysters on the half shell. Usually it is make with white pepper, but I changed it to black pepper for a variation.
¼ cup finely shallots (about 2½ ounces)
¼ cup champagne vinegar
1/8 teaspoon sugar
1/8 teaspoon salt
1¼ teaspoons crushed black pepper (Do not use ground pepper. Crush peppercorns with the back of a large knife.)
Bloody Mary Sauce
Here is a bright, colorful sauce with a little kick, using some of the ingredients from the traditional drink.
¼ cup tomato juice
1 teaspoon horseradish
½ teaspoon freshly minced dill
Dash or 2 of tabasco sauce
Dash of Worcestershire sauce
A few shakes celery salt
For a sweet and sour sauce, this one brings the two flavors together.
½ cup rice wine vinegar
1 teaspoon peeled and finely grated ginger
½ teaspoon sugar
1 teaspoon tamari or soy sauce
1 teaspoon sesame oil
½ tablespoon fresh lime juice
1 tablespoon minced chives or scallion
For each recipe, combine all ingredients in a small bowl and let rest for at least 30 minutes to an hour in refrigerator for flavors to meld. Serve chilled with small spoon for serving.