As far as I can tell, there are no classic or legendary cocktails named after Memphis. Aside from a few novelties that have shown up in various databases, I haven't uncovered anything serious or notable bearing the name of our fair city. By contrast, there are separate cocktails bearing the names of New York's five boroughs. The
Manhattan is perhaps the most famous and most enduring, and has become a favorite of mine over the past few years. But what of the other four?

The
Queens Cocktail is a perfect martini with pineapple (perfect meaning equal parts red and white vermouth).
The Queens Cocktail1½ oz. Gin
½ oz. Sweet Red Vermouth
½ oz. White Vermouth
1 Ring of Pineapple
If you're using canned pineapple, stick to the stuff packed in juice, not syrup. Or to make it faster and easier, consider substituting half an ounce of pineapple juice instead of working with the fruit. Muddle the pineapple in the bottom of a cocktail shaker. Add the remaining ingredients, add ice, and shake and strain into a martini glass.
As I said, this is basically a martini, but the sweetness and fruit juice give it a softer edge, and it's better suited to hotter weather.

The
Bronx Cocktail is similar to The Queens in that it is a modified perfect martini, but with a bigger portion of fruit juice. According to some sources, if you add Angostura bitters this becomes an Income Tax Cocktail, though like most recipes from the turn of the last century there are many variations.
The Bronx Cocktail2 oz. Gin
1 oz. Orange Juice
½ oz. Sweet Red Vermouth
½ oz. Dry White Vermouth
Combine ingredients in a shaker with ice. Shake and strain into a martini glass.
You get the impression that this was a morning eye-opener for folks in the pre-Prohibition era, a beverage that, like the Bloody Mary, could be rationalized as part of a healthy breakfast. It's actually softer and better balanced than a Screwdriver, even though the gin and vermouths add so many other elements to the drink.

I had a weird experience with the
Staten Island Ferry. The drink, not the water transport. For the latter, my grandfather has pictures from a Navy shore leave circa 1946 where he and a buddy are hanging out with a couple of young women from Staten Island. He pointed out that only because the ladies gave them fake phone numbers would he later meet my grandmother, eventually leading to my birth 30 years later. So, er, thanks Staten Island gals!
In my own odd story, a girlfriend who didn't drink much tried to order a cocktail in a bar in Dallas. She told the bartender that it was made up of Midori melon liqueur and Kahlua. He politely suggested that wouldn't work well, and she and went through a few other iterations before eventually remembering that the combination she was thinking of was coconut rum and pineapple juice. Without further ado...
The Staten Island Ferry1 oz. Coconut Rum
1 oz. Pineapple Juice
Combine in a highball glass with ice, and serve. It is precisely what it looks like: a quick shooter that is a lazy variation on the Piña Colada. Too sweet for my tastes, and also a much newer entry to the cocktail world. (The other three recipes shown here have been around since around the Prohibition era.) But there are a lot of people who would really enjoy this.

I saved my favorite for last. The
Brooklyn Cocktail is a clever, delicious, and classy variation on the Manhattan.
The Brooklyn Cocktail1½ oz. Rye Whiskey or Bourbon
½ oz. Dry White Vermouth
¼ oz. Maraschino Liqueur
¼ oz. Amer Picon
What's Amer Picon? It's a form of bitters from France. It's fairly obscure these days, so I substituted Averna Amaro. Combine everything in a shaker with ice, shake, strain into a martini glass and serve.
The baseline of this cocktail is very much like the Manhattan, but on top of that you get all the nice herbal notes from the bitters and the nutty, cherry flavors from the Maraschino.