Bartenders can often come up with a new drink and claim it as their own, but copyright laws in the United States put a chill on legally recognizing that claim.
Recipes Not Protected
Copyright law explicitly does not recognize recipes that consist of a listing of ingredients. It can protect the creative description on how those ingredients are used, but not basic instructions such as "shaken, not stirred."
How about the name?
Titles and names are not protected under copyright laws, but other legal options exist.
If a bar publicizes a drink by a unique name, the name can be trademarked by the U.S.Patent and Trademark Office. That doesn't stop someone else from making the drink and calling it something different.
Marketing the Name
If your drink becomes a trademark of a particular bar or restaurant, you can restrict other places from using the name. From a marketing standpoint, it suggests the only place to get the original is at the place that owns the name.
In a Book
If your drink gets picked up by an author of a cookbook or bartender's guide, your actual description of the drink falls under copyright protection of that book, and you may receive fees as part of the deal.