I've heard that Annie Proulx finds many of her idiosyncratic character names from cemetery tombstones. It's a question I don't often see asked of writers - how do they come up with character names? I think it's an important question because the right name can work so well for a character - think of Thomas Covenant in Stephen R. Donaldson's books. Of course the wrong name can not work so well - I confess I giggle every time I think the name "Rayford Steele" in the Left Behind series.
As for my own naming process, I generally try to avoid "loaded" names - those can work well, as in the aforementioned Thomas Covenant or Vonnegut's Billy Pilgrim. But in the wrong hands they can be deadly and I'm not up for the risk. Probably the closest I've come to a loaded name was giving a female sociopath the name Juliette - I named her for De Sade's Juliette, not Shakespeare's Juliet.
Often I get a mental image of a character, and think a bit about the character's situation when the story starts, as well as about his or her past. For first names, I start "trying on" character names to see what fits. Baby name web sites are quite helpful for this, especially if I'm looking for an unusual name or one that fits a particular ethnic heritage. This leads us to last names - when I know a bit about the character I have an idea of his or her family background and from there will look up surnames relating to that character's heritage, whether it's Irish (Cahill, Bannion, Kincaid, Monahan), WASP (Halsey), Armenian (Danayan), Czech (Novak), French (Beaumont, Delacroix, Pavour), Portuguese (Salto), German (Kessler), or relatively generic (Thomson, Whitman, Finley, Cross).
I don't think there's a writer alive who doesn't love certain words just for the way they sound (I'm strangely fond of the word "congeal"), and names are no different. I have a mental list of names, both first and last, that I just like the sound of. I named a character Evie for a girl I knew in college - I'd always liked the way her name sounded and was pleased when I found a character to give it to. Likewise the last name Kincaid, which just had a nice ring to it.
A good source for surnames is your workplace's employee directory (it helps if you work for a big corporation). I needed an ordinary name for an ordinary character and had the first name (Jennifer) but couldn't think of a surname. After trolling through the directory for commonplace names, I found the name I needed (Thomson - and it was a weird feeling when I one day ran into a person named Jennifer Thomson. Thankfully she looked nothing like my character or I'd have been really creeped out.)
Occasionally I'll take a name from fiction - I gave a character the last name Tally after reading Ray Bradbury's haunting story "The Lake", for the lost and drowned girl the narrator mourns for. Another character's surname, MacReady, came from the name of Kurt Russell's character in The Thing.
And I also like to name incidental characters (what my Mom calls "the milkman who arrived on the big day" characters) after writers I like. So far I've taken surnames from Annie Proulx, Richard Yates, Anna Quindlen, and more I can't recall now. There are too many names - I can't remember them all! I need to start keeping a database!