The drink he does order is a Vodka Martini, famously requested as "shaken, not stirred".
It's not a specific cocktail with a unique name, but here's the standard recipe:
* 2 oz Vodka
* 1 oz Dry Vermouth
* Lemon peel (for garnish)
Instructions:
1. Fill a mixing glass with ice.
2. Add vodka and vermouth.
3. Stir vigorously for 15-20 seconds.
4. Strain into a chilled martini glass.
5. Garnish with lemon peel.
While there's no "official" Bond Martini recipe, this is the general way it's prepared.