Who are some important people who have had an impact on society and are known only by their first names?
The most famous people who are known by only one name are people who didn't have last names at all. These are people from ancient times such as Moses or, more historically, Aristotle, Plato, Solon, Homer, or Pythagoras.
Later on, in Roman times, people did have last names but we essentially never use them. From that era, there are people like Cicero and Nero. In term of people who made a great impact on society, I suppose that Constantine from that era would be high on the list since he made Christianity the official religion of Rome. On that subject, Paul was a very important person known only by his first name given that he was one of the people most responsible for the spread of Christianity.
In more modern times, there were plenty of monarchs known by first names and titles. These include Elizabeth I and Louis XIV.