So, now that you have your (probably) long list of favorite names, here are some ideas on how to start organizing and minimizing your names.
Taking the example of just searching for one particular character, I'd suggest organizing the names into alphabetical order. Separate the first, middle, and last names either into 3 separate sheets of paper (or 3 piles of index cards, whatever you used) or, if you used a program like Stickies on your computer, sort everything into 3 sticky notes.
Go down through your first names and read them out loud; whichever name doesn't immediately jump out at you or at least give your mind a little tug, eliminate. (Tip: Don't get rid of that name forever. Get it out of sight, but save it in another document or on another piece of paper. Later on if no other names work out, come back to it, or save it for future characters.) Chances are, if you don't feel a tug towards the name right away, you won't later.
--> If you have a picture you use as inspiration for your character, now is the time to keep it nearby. As you read off names you can see which ones look and sound best with your character.
Go down through your "revised" list several times, trying different middle and surnames with each name. Don't get discouraged if your favorite name doesn't sound right with the surnames you picked. You can probably work around it by at least choosing a different surname (or perhaps you might discover having a surname isn't all that important to the story!).
At this point you might find yourself with two or more names that you think are perfect, or at least close to it. Don't stress; in fact, it's probably better that you do, because the elimination process isn't necessarily over. There's still a lot to consider in deciding on the perfect name.
However, you may have one name that you keep thinking about. If so, take that one and, if you have any other character names picked out for the same story, make a "cast" list. You'll want to make sure that you don't have all your names starting with the same initials or containing the exact same amount of syllables. You might find that right there will help you determine which name from your minimized list is the best.
One last step in this process: google your final contestants. You may not realize it, but that brilliant name you just pieced together might be the name of a famous scientist, or the name of an already existent character. Just as an example, I've been toying with a story idea for a few months now. I used many of these processes I'm discussing with you in creating the male lead's name, and finally decided on Aaron Palmer. I adored the name, but something wasn't setting right. It sounded so...familiar. A month after I chose the name, I was in the store and saw a can of Arnold Palmer tea. It had taken me forever to make the connection consciously, but obviously something in the back of mind was trying to force me to use a name I'd already heard before. Definitely be cautious in this area; you don't want people to think you're copying.
In part 3 we'll close with discussing if "significant" names are important to a story.