Note that despite the name, this is a classical French sauce, not a Russian recipe.
There are two versions of Russian sauce in classical French cooking.
While it is clear that Russian Sauce 2 might take its name from the use of caviar, it’s unclear where version 1 might get its name from.
Russian Sauce 1
(Primarily for beef)
1 tablespoon each of chopped parsley, chervil, tarragon. After chopping and measuring, blanch and drain them;
Mix into some thinned velouté sauce;
Add 1 tablespoon mustard, ½ tablespoon sugar, dash of ground pepper, and juice from 1 lemon.
Russian Sauce 2
This would be a creamy, pinkish sauce owing to the use of the red lobster coral (egg mass.)
3 parts mayonnaise
1 part of the following mixture: caviar and lobster coral, sieved
Flavour to taste with mustard
Literature & Lore
Fannie Farmer gave a recipe for something she called Russian Sauce, to be used with beef. Her recipe calls for mustard and lemon as does Russian Sauce 1 above, but otherwise, it bears no relation to either of the two versions recognized in the French cooking canon; the Russian allusion likely comes from the use of horseradish and cream.
Russian Sauce
3 tablespoons butter
½ teaspoon finely chopped chives
2 tablespoons flour
½ teaspoon made mustard
1 cup White Stock III [Ed: plain chicken broth]
1 teaspoon grated horseradish
¼ teaspoon salt
¼ cup cream
Few grains pepper
1 teaspoon lemon juice
Melt butter, add flour, and pour on gradually White Stock; then add salt, pepper, mustard, chives, and horseradish. Cook two minutes, strain, add cream and lemon juice. Reheat before serving. Serve with Beef Tenderloins or Hamburg Steaks.
— Fannie Merritt Farmer. The Boston Cooking-School Cook Book (1918 Edition): Fish and Meat Sauces. Boston: Little, Brown. 1918
Sources
New Larousse Gastronomique. Paris: Librairie Larousse. English edition 1977.