I really enjoy the process of naming my characters. I think a lot about name meanings and research the history of the names before I finally settle on anything for a character. There are a lot of great sites out there for just that purpose.
Sites I like for character names:
(they even do last names - bonus!) http://surnames.behindthename.com/
superhero names: http://www.seventhsanctum.com/generate.php?Genname=superheronameorg
One thing that I really like to do is to read movie credits for people's names - I find it's a great way to get a variety of both first and last names that you might not come across otherwise. Plus you probably strengthen your eyeballs by trying to read all those names as they zip by.
I find it interesting when people just pick names randomly for characters, given how much time I put into naming mine - must be easy to just go with a gut feeling and stick with it.
Given all this time I've spent on researching character names, you'd think picking a baby name would be easy, right? Wrong. Apparently I have to take my husband's opinion into consideration. Weird. And, if I change my mind down the road, I'm apparently not allowed to do a "replace all" in the manuscript and just change the kid's name willy-nilly.
This is why fiction is better than reality :)
What methods do you use to pick a character name?