For the Jay & Jack podcast about "the Life and Death of Jeremy Bentham" they postulated that a "Kate on the Rocks" might be a mixed drink consisting of whiskey on ice with a dash of sweet & sour. I'd garnish it with something that might represent freckles, like nutmeg or cinnamon, but some might think that over the top. The thing is though, you could take drinks that already exist and make them Lost drinks, by changing the garnish.
Just off the top of my head, a Hurley could be a Tequila Fizz (Tequila, Lemon juice, Grenadine, Egg white, and Ginger ale) garnished with two olives, a pineapple wedge, and a golf tee - maybe cherries after that if there's still room. You should have to eat the garnishes to get to the drink. A Cursed Hurley could be without the tequila. An Incarcerated Hurley could be twice the tequila and without the ginger ale. If you order a Hurley "by the numbers" it comes with a burger and fries. A Hurley Dude could be without the egg white, and just the golf tee.
A Jin Sun could be like a Saketini (gin & sake) but with a shrimp instead of an olive.
A Sawyer is just a beer, but it wouldn't be a drink by itself. You could have for example a Kate On The Rocks With A Sawyer, which would be whiskey on ice w/sweet & sour... and a beer chaser. You can have any other Lost drink, with a Sawyer, and you're basically asking for that drink and a beer.
A Jack Shephard could perhaps be equal parts Dr Pepper and Jack Daniels on the rocks with a dash of lime juice in a respectable glass. No garnish, but this would go great with a Sawyer.
A Desmond Hume might be two shots of expensive scotch whiskey, one shot of cheap brandy, a dash of vermouth, and a lime peel twist. In a collins or highball glass.
If anyone can think of other possible Lost Drinks, please feel free to share your ideas. I don't know if this could someday become a Lostpedia article but you never know. Crazier things have happened.