See also: 4A40.34 in Chemistry and Physics of Everyday Experience 

Ingredients:

• 6 cups of cream/half-and-half/milk, in a 2:4 ratio of cream to half-and-half
• 2 cups of sugar
• 1 tablespoon of vanilla extract

Optional Ingredients: 

• 1/2 teaspoon lemon extract
• fruit chips
• chocolate syrup if desired

Now to freeze it....

Individual servings can be made quite easily.  Give each person a Styrofoam cup that is half full of the ice cream mixture.  Pour a small amount of liquid nitrogen into each cup while the person stirs until it freezes.  In this way each person gets to play with the liquid nitrogen and make their own ice cream. 

OR.........

Get a "large" metal bowl and a wooden (or other sort of material that won't break when cooled to very low temperatures) spoon and pour the mixture in.  Add ~ 1-2 liters of the LN2 and stir vigorously.  (This will be very impressive to the kids as it will look like you're stirring a cauldron of witch's brew!)  

CAUTION: Adding the liquid nitrogen too fast or without stirring will result in ice cream with large lumps or mixture inconsistencies that will affect the taste.  

If you have something to add to the ice cream (fruit, nuts, etc.) get the ice cream cold enough until it just begins to lose it's shiny appearance, then add.  From here just keep adding the LN2 until the ice cream is the desired consistency.  Serve immediately!

You can get ideas for flavors and items to add by looking in a cook book that has regular hand-churn ice cream recipes and adapting them to his recipe/freezing method.