Combine sugar, cranberries, dates and eggs in a medium saucepan. Cook over medium-low heat, stirring constantly, until the mixture thickens, is pale yellow in color, registers at least 170°F on an instant-read thermometer and when a spoon is pulled through it, it leaves a clear trail. This will take 6 to 14 minutes, depending on how hot your stove's "medium-low" setting is.
Remove from the heat; stir in pistachios, walnuts, pecans and rum extract. Let stand until cool enough to handle, about 45 minutes.
With damp or lightly oiled hands, form the mixture into 1-inch balls (about 1 generous teaspoon each). Roll each ball in coconut. Place the finished balls on a baking sheet lined with wax paper and store in the refrigerator.