Everything, or least to me. I spend a lot of time finding just the right name for the characters in my stories. The wonderful thing about naming a character, as opposed to naming a baby, is you know the character's personality before settling on a name so you're able to pick one that perfectly sums up the character for a perceptive reader. The right name can be a clue, or a hidden joke.
I even collect names. I have a journal devoted to names that I find and like. In it are first names, last names, full names, names from different eras because names go in and out of fashion. Don't believe me? Give it fifty years and Brittany will seem as outdated then as Gertrude does to our ears. I find names just about everywhere. While reading (history books are especially good hunting grounds), in those baby books filled with lists of names, phone books, graveyards, even the waiting room at the doctor's has given me some treasures.
Try keeping lists of interesting names for yourself, if you don't already. Sometimes just a name can be so inspiring a character will grow out of it. If you'd like, you can turn this into a Writing Adventure Exercise. A great place to visit is city archives, or a small local museum - any place where you will find plenty of photos of people from long ago. Read the names of the strangers in those old black and white pictures and I guarantee you will find something to inspire you. Before you know it you will have a list of fantastic names and from them characters will come alive in your imagination.