Boil 3/4 cup water in small saucepan. Remove from heat. Add 4 green tea bags. Rip open an additional tea bag, and add tea leaves to water. Let stand 6 minutes. Remove bags from the pan, squeezing any liquid into pan; discard bags. Return pan to medium-high heat. Add 2/3 cup sugar; stir until sugar dissolves. Remove from heat, and cool (about 15 to 20 minutes).
Combine tea mixture, 4 cups cubed honeydew melon, 6 tablespoons melon-flavored liqueur (such as Midori), and 2 tablespoons fresh lemon juice in a food processor; process until smooth. Pour into 11- x 7-inch baking dish. Freeze 30 minutes or until edges are frozen and center is slushy. Stir granita with a fork, scraping frozen edges into the center; freeze 20 more minutes. Repeat this step every 20 minutes until completely frozen into light, icy crystals (about 2-2 1/2 hours). Scrape into 6 glasses to serve.