Inc.'s Glenn Leibowitz has written a good article with three tips for delivering an impactful elevator pitch.
First, he argues that you must create three different versions (one-liner, 30-second nugget, and 60-second double-click). I think it's good advice for two reasons. You will have different occasions at which to deliver your pitch, and you may want to draw on different versions for these contrasting occasions. Moreover, boiling a 60-second pitch down to a shorter version forces you to think critically about what is absolutely the essence of your argument.
Second, Leibowitz explains that you should tell a story. He says, "You are the protagonist --the hero -- of your own story, and you're on a mission to achieve a goal: "Build a new business", "grow an existing business", "move into a position of greater responsibility." And, like any good hero, you probably face challenges and obstacles that you'll need to overcome if you hope to achieve your goal.
Finally, Leibowitz argues that you should practice the pitch on a friend, and ask them to play back what they have heard. He explains that you may get some good ideas from them about how to deliver the pitch, since they may choose different words and phrases to communicate your ideas. I would add another reason for why taking this approach can be effective. You may hear back something quite different than what you meant to say. That unexpected response from your friend will tell you that you need to clarify your pitch.