"edit #2: went to that site Kassios referred to and, unfortunately, I'm not a student at Oxford so it won't let me search for names! Is there another reliable website with modern AND ancient Greek names that I can use?"
Click here for ancient Greek names: http://www.lgpn.ox.ac.uk/publications/index.html
and then click on LGPN I, LGPN II, etc. and then to the "forward" or "reverse" index PDF's and you shall have thousands of Greek names.
Unfortunately you do need to be a student or buy the books to have more detailed information on the names...