I am with an organization that has members around the world; most in
North America and Europe. I would like to plot their locations and
then find out the center of the locations. This will help decide where
to hold our convention (shortest distance for the most members). The
ideal solution would be software where I click on a map to locate
members then hit a button to find the geographic center. Second best
would be a way to specify cities. I do not want to enter GPS-type
coordinates since this would take too long to look up for each member.
Cost is also a factor. A free or low-cost solution is preferred. I am
only doing this one time so I wouldn't pay to buy dedicated,
professional-level software.

A no-cost alternative could be labor intensive on your part (whether
you do it and the number of members).  Whichever continent has the
most members should hold the convention.  Find all the convention
centers that are capable of handling the number of members you have. 
I would start with the most central convention center based on its
address.  Put the convention center's address in Google maps,
Mapquest, etc. and the member's address.  That should tell you the
distance in miles.  Record that in a spreadsheet.  Use the spreadsheet
to calculate the average.  Compare the average to all the other
convention center averages.  The lowest wins.

For a one-off exercise, I would keep it simple.  If you can sort your
membership address list by cities, I would get a count of the larger
groupings, put the numbers on the places on the maps (Europe and
N.America), and just eye-ball the results.  Keeping in mind that air
travel convenience is also important, you are going to have to
compromise on the actual geographic center, so a more precise effort
would probably be more trouble than it is worth.  The conference
should not only be close to the most people, but also easily
accessable by all, preferably non-stop or no-transfer flights.