From the get-go in your career, even as a law student, you have to differentiate yourself. One tool for that is your elevator speech. You bet, you need one. Now.
Essentially your elevator speech is the short-hand introduction of yourself to the world of business. Law is entirely a business, no longer a profession.
So, you never will introduce yourself as "Joe, a student at Columbia Law School." That's generic. That screams you are heading in no clear direction. Laser-like focus is demanded.
Instead, you say, "I'm Joe Jones. At Columbia Law School I am working with several professors on simplifying briefs - or how lawyers communicate with the courts. And you?"
Your elevator speech is not cast in concrete. You change. The value of specializations changes. The world changes. So will your speech. But you will always keep it brief. And it will always, Dale Carnegie fashion, lead back to a sincere interest in the other person.