Keep extra clementines on hand in case some of the skins tear and for juicing. If you don't feel like juicing enough clementines to yield 2 cups, use store-bought orange juice to make up the difference.
Slice the top 1/2 inch off each clementine; reserve. Cut around flesh and scoop out into a sieve set over a bowl; reserve skins and be careful not to tear them. Press flesh to extract juice. Squeeze in juice from tops. (You'll have about 1 cup.) Juice more clementines to yield 2 cups.
Moisten clementine skins with some juice or water. Roll skins in 1/4 cup sugar to coat. Freeze until solid, about 2 hours.
Bring remaining 1/4 cup sugar, 1/4 cup water, and ginger to a boil in a saucepan, stirring, until sugar dissolves. Remove from heat. Let stand for 30 minutes; discard ginger. Stir in 2 cups juice and the lemon juice; transfer to a nonreactive 8-inch square baking dish. Freeze until solid, about 3 hours.
Scrape granita with a fork to fluff. Spoon granita into clementine "cups." Freeze for 10 minutes before serving.
Granita can be frozen for up to 2 days. "Cups" can be frozen for up to 2 days (unfilled) or 1 day (filled).