Thanks for the tip. The dual pinging (to the Router & my ISP's IPs) showed the same results.
It turns out that my Lion-equipped MacBook and Mac Pro loses the Internet connection about once each day, during periods of inactivity. I have pulled the Ethernet cable from the Mac Pro to connect via WiFI. When the Internet lose occurs on the MacBook, turning WiFi off and on again re=establishes the connection. Not so with the Mac Pro ... the big machine needs to be restarted to regain connectivity.
Somehow, I became worthy of having Apple assign me an Engineer and a Case Number. We talk and email each other about 3-4 times a week as I try various configurations and activities. Apple has send me a couple of Apps to run and log data to assist in finding out waht's going on. The most recent App ("LogMan.app") was only written by Apple last week. It generates a text log file that can be forwarded to Apple for analysis.
So while the WiFi and Internet dropping still occurs with Lion, the good news is it seems that Apple is working to locate the bug in 10.7.3. Since they haven't found it yet, we can assume Mountain Lion will have the same issue.