Oysters on the Half Shell
Here are three suggestions, all slightly different.

Mignonette Sauce
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.
Ingredients:
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.
Ingredients:
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
Asian Sauce
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
Ingredients:
½ teaspoon sugar
1 teaspoon tamari or soy sauce
1 teaspoon sesame oil
½ tablespoon fresh lime juice
1 tablespoon minced chives or scallion
Directions:
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.