It seems that pasting the following directive into the site's "Apache Directives" box resolves the issue:
The directive value, 5000, is equal to the value that is defined in System -> Server Config -> example.com -> FastCGI [tab] -> FastCGI max. Requests.
Given that this problem emerges in a default ISPConfig installation, my humble opinion is that this should be fixed in ISPConfig. That is to say, ISPConfig should automatically add "FcgidMaxRequestsPerProcess 5000" to each site's virtual host configuration (where 5000 is the value defined for "FcgidMaxRequests", via the interface):
# FcgidMaxProcesses 1000
Better yet would be to add another field, "FastCGI max. Requests per Process
", to the FastCGI configuration interface, so that this value may be controlled independently of "FcgidMaxRequestsPerProcess".
I have added a thorough explanation of this problem to the relevant ServerFault question at http://serverfault.com/a/442768/142864