After having one of these at a local Indian restaurant, I investigated how to make one myself. Unfortunately, I was daunted by the total lack of lychees or their syrup at ordinary grocery stores. My compatriot, however, proved my stupidity in the matter by locating canned lychees at our World-Famous (tm) Dekalb Farmer's Market. Maybe they'll get proper fresh ones when they're in season.
Anyway, my mad science mixological experiment turned out to be rousing, sousing success, so I present my recipe to you, Dear Internet Reader. Though not strictly speaking a "traditional" gin martini, I figure I can get away with calling it such since I actually use vermouth.
Remember, enjoy responsibly: always drink while wearing a condom.
Lychee Vodka Martini
3 parts vodka
1 part dry vermouth
1 part peach schnapps 
3 parts lychee syrup (recipe follows)
1-2 lychee fruit
Mix over ice in martini shaker. Shake until mixed and cooled. Serve in a martini glass with lychee fruit as garnish.
 Many other kinds of schnapps probably work, too. Triple sec could also work well. Go forth and experiment!
0.5 cups water
0.5 cups sugar
8 canned lychee fruit
0.5 cup syrup, reserved from canned lychees
1 teaspoon lemon or lime juice
Combine sugar, lemon/lime juice, and water in a small saucepan and heat until boiling. Continue boiling for approximately 2 minutes. Set in refrigerator to cool.
Once the simple syrup is cool, mix in blender with reserved canned lychee syrup and the lychee fruit. Blend until smooth.
To remove most of the lychee pulp, use a fine strainer. Refrigerate before use. May be stored for up to two weeks in the fridge.