Well, two of my works (and a possible idea for a third?) are at least vaguely based off Greek mythology. Of course, then I don't have much of a choice in what my characters' names are.
In other things, I get to choose (which is more fun!). Sometimes, the names come with the plot. Steam, an invention from a dream, had some names pre-made. If the character is nameless, I do the lamest thing you can possibly imagine, and that is to go through an online list of names. Depending on the culture/setting of the book, I'll find a list of corresponding foreign names. I really like French names, although curiously, I have no French-named characters. I pick names based on how they sound. If I like how it sounds, it's a winner. Special significance is nice, but if I have a character that is associated with the Water element, I will not go out of my way for a name meaning "water" or some such.
Another important part: connotation. Some names already have a connotation, and I don't want to mess with that. Zanna and Deeba are funky type names (Un Lun Dun). If I have a dorky, awkward character, I need to pick a dorky, awkward name. If you run too much against a name's connotation, you run the risk of giving readers a preconditioned idea of the character's personality. For example, before I read Percy Jackson, I despised the name Percy. Seriously, do you know any kid with that name? The only kid I know is Percy from Harry Potter, and he's a total stuffy douchenozzle. I didn't want to read a book about a kid who was a stuffy douchenozzle, so I didn't. For at least a year. My brother mercilessly reminds me it was he who recommended the series first, and I apparently told him it looked stupid. That's an extreme case where a weird name completely turned me off from a book. Of course, in the book, it is explained Percy is a shortened form of Perseus. This makes sense. But Perseus Jackson is also a pretty crappy name for a book.
To avoid this whole problem, I try to find names that are unique. Some names I've used: Simeon, Nic, Elladora, Niles. These are not extremely popular names currently. I intersperse some common names: Sarah, George. This is so the whole book doesn't come chock full of eccentric names like I'm trying too hard to be artsy and original. Furthermore, I picked Sarah and George because there are so many people and characters who have those names, it's hard to attach a preconceived personality to them.
Finally, I make an effort to pick names that do not sound similar (this is a general rule of thumb for most writers) to avoid confusion when reading. At the very least, I pick names that don't start with the same letter. Unless one of the characters dies way early or something.
Naming a character is like birthing a new person into the world. Once someone has a name, he/she has a life. Names are important.
Be sure to check out everybody else's naming process!
Wednesday: Lin Wang - Teen Writer
Thursday: Girl With A Notebook
Friday: Somewhere Nowhere In My Kingdom
Unrelated question: I am revising ATRS right now, and it's horrible. Rewriting is huge, but there are so many problems I don't even know how I should approach this. I could either go through front to back and rewrite that way, or pick out really, really bad scenes and redo them. But I'm also trying to work in some clarifications to the plot, fix some unconvincing motivations, etc. How do you all prefer to rewrite? Any kind of help is appreciated.