What a great question.
While I can't claim to have tried one everywhere, or even many, I have tried a few around the World. You have to decide if you want atmosphere, too.
My favorite, at the Jazz Club at PI, is no longer available. Atlantic Dance at boardwalk used to have a dedicated martini menu, but they may have done away with that when they stopped being a swing-dance place. (And it seems like they're barely open at all anymore.)
One choice is at the Outer Rim at the Contemporary, but depending on what time you go it can be a madhouse... it's right next to the steakhouse, and there's a big waiting overflow. Late afternoon seems to be a good time, before everyone returns from the parks.
Kimonos in the Swan isn't bad, either. And you can have some sushi.
Haven't tried Mizners at the GF, but I suspect it would be good.
Since this is a first time thing, make sure you ask for the good stuff.
(Bombay Sapphire is my favorite.) And if you like it and feel like another and really like olives, try a dirty martini, mmmm.