Is "/private" just an ordinary directory or does it have special properties The nominal links
 /etc --> /private/etc,
 /tmp --> /private/tmp and
 /var --> /private/var 
are all straightforward and I understand they are there because of a legacy need that has passed. Nothing magical about them.

However, I just saw these install instructions for an Apache/httpd module which recommend using /private/libexec instead of /usr/libexec. That directory doesn't exist but I can create it via sudo. So, I am wondering if /private has some special properties such that /private/libexec is merged with /usr/libexec?
There is no magic at works here, this will just create a user-writable directory to store the module in. You can‘t copy it into /usr/libexec/apache2 as this is protected by SIP. 

Using /private like this seems kind of odd though. I would probably create /usr/local/libexec/apache2 and install the module there. You just need to be careful to adapt all references to it as well.