Are there any FREE websites available to locate someone &
what are the best websites which require a reasonable fee for this
type of search.  I live in NJ, my job is following up on patients who
were in clinical trials to keep track of their long-term results. 
Many have died or moved.  Some of the medical charts are outdated
where there isn't sufficient info (SS, Birth).  In some cases I only
have the last known city/state.

Since I'm not a Answer guy, i'll give you the needed information
anyways (because i'm board right now).  There is NO free way to locate
someone using online webpages.  If your cheap (I am) you'll need to
travel to the local records (courthouse) in your area and manually
search for the person; however, if you wish to pay a small fee ranging
from 7.95 to 49.95 where each level gives additional information.  The
website I recommend (and the cheapest I found) is :

https://find.intelius.com/search-name.php?searchform=name

I hope this information is helpful.

You may also want to try www.searchsystems.net.  It is a free website
that provides many useful links broken out by state.  Another
suggestion is to verify deaths through the social security death index
which is available through a variety of websites (a quick search
"social security death index")will point you to specific websites.
With the SSDI, you can search on a combination of fields, so you don't
necessarily have to have the SS#. I believe LexisNexis, Accurint, &
KnowX may have some databases that you could utilize for a fee as
well. Since you are working with patients, you may have to ensure that
using a fee-based service is in compliance with HIPAA.

give http://www.zabasearch.com a try

There is a free way of doing this, but it is a bit dishonest.  You can
join several large free email services that allow users to access
personal information on other users.  I know that hotmail does this
(an old girlfriend found me this way.)  It is, in my opinion, a
violation of privacy, but any idiot willing to post private info on
public forums deserves whatever they get (as in an old, kooky
girlfriend finding you).

Try www.abika.com. They are another fee based people locator service.