I won’t quibble with anyone who prefers the lemon twist in their martini, but to me, the olive is always the best part of a martini. I don’t take mine dirty (except occasionally for research purposes), but still, in my mind, the martini is just a plain glass of gin and dry wine without the olive.
To clarify, not just any olive will do. You’re looking for Spanish (green) olives. The standard cocktail olive is usually stuffed with a little piece of pimiento. While they come in all sorts of sizes, people usually like the large (queen)-sized olives. The flavor is the same, but this is a garnish we’re talking about, and the garnish is all about the visual appeal. (If you’re not using large olives, you’ll want to put two or three on a toothpick.)
Depending on your taste, you can branch out to specialty olives. Olives stuffed with jalapeños are a bit more spicy, obviously, but if that’s your thing, it can be a nice palate-cleanser. Same for garlic-stuffed, anchovy-stuffed, tomato-stuffed… none of it changes the flavor of the martini, but they do add a bit to the finish.
One exception is the blue-cheese stuffed olive, which will definitely impart a rich flavor to a martini. I especially like to garnish a smoky martini (one with just a dash of a nice blended Scotch like Johnny Walker Black) with a blue-cheese stuffed olive. It’s a nice, albeit rich, combination.
While we’re talking about olives, we might as well remind you of our discussion of cocktail onions, which are used to garnish a Gibson cocktail. (Or a glass of water intended to look like a martini, and garnished with the onion so that the drinker would know which one was the fake martini.)