Thanks, pa28pilot! That's consistent with what I've come up with after doing a lot of research and reading to try to educate myself.
This is a problem at least in that the volume and frequency of the DNS server error messages make it near impossible to find any other, possibly significant, error messages: needle in a haystack effect.
Recently, for no reason apparent to me, the file-not-found messages stopped, and the following kinds of messages are appearing over and over in the system log:
Feb 17 21:29:15 2016 named[1251] err<139> zone localrp/IN/internal-clients: zone serial (2008122601) unchanged. zone may fail to transfer to slaves.
Feb 17 21:29:15 2016 named[1251] err<139> zone fios-router.home/IN/internal-clients: zone serial (2008122601) unchanged. zone may fail to transfer to slaves.
Feb 17 21:29:15 2016 named[1251] err<139> zone FIOS_Quantum_Gateway/IN/internal-clients: zone serial (2008122601) unchanged. zone may fail to transfer to slaves.
Feb 17 21:29:15 2016 named[1251] err<139> zone 1.168.192.in-addr.arpa/IN/internal-clients: zone serial (2008122601) unchanged. zone may fail to transfer to slaves.
If I did want to stop the DNS server errors, the only thing I can think to do would be to use SSH to log in to the router and disable the named (BIND9) service. Does that make any sense, or is it better to just ignore the errors?