Sometimes a name will spring into my head as soon as a character starts talking to me. But here are some of my sources for inspiration when that doesn't happen.
Character names are super important to me, and I always try to get them set before I even start in with the serious writing of the play. What a John might say is so different from what a guy named Patton or a guy named Francis or a guy named Sue might say!
One thing that has helped me is the profusion of baby name sites on the web. Not just American ones, either! Is your character Cambodian? Chinese American? Bolivian? Just google baby names and the country of choice, and you'll get all kinds of possibilities.
Another nice tool I found somewhere online was a government site that listed common-ness of names by birth year. So if your grandmother character was born in 1938, you can see what names she might have had, statistically speaking. More likely a Delores than a Brittney! Very helpful!
I’ve also been known to skim whatever books I’m near and use various combinations of authors names in a pinch.
How do you name your characters?