How to use herb extracts or essential oils in drinks
Sugar will bind with the essential oil, allowing you to produce a flavored syrup that you can use to sweeten the drink in place of sugar. This is a commonly used practice in drink-making since a syrup is much quicker to integrate than dry sugar.

For a typical simple syrup recipe, you combine equal parts water and sugar and heat them until the sugar dissolves. With a flavor extract, I typically add the extract to the sugar first, let the sugar absorb the extract, then add the water and proceed as normal. You'll need to experiment with the amount of extract to achieve your desired level of flavor. 

Keep the resulting syrup in an airtight container in the refrigerator; it will keep for at least a couple weeks. If you increase to two parts sugar to one part water (sometimes called "rich simple syrup") or add a small amount of alcohol, you'll get much longer shelf life.

Your other option is to bind the oil with some other emulsifier; egg white is commonly used in cocktails for its ability to form stable, thick foams (which is just an emulsification of air in liquid). The disadvantages with using egg white are that it's difficult to measure in small portions, it will change the texture and flavor of your drink somewhat, and you'll need to agitate (i.e. shake) the drink in order to emulsify the proteins in the egg.