Alright - now we're getting somewhere!!! I noticed that monitoring my Ethernet utilization at the time the freeze happens there is a large spike in in Receive utilization, and a large decrease in Send. Disconnecting the ethernet cable the problem goes away, reconnecting it comes back.
Don't even know how to begin to troubleshoot that. Baseline Send is around 4 Mbs, Receive is close to zero. The drop off from Send is down to 1 Mbs, the Receive goes up to maybe 1 Mbs. Neither of these are huge numbers so it's not that the Ethernet card is getting pegged. But regardless the pattern is indicative of the problem. Unfortunately this computer does not have a wifi card so I can't check to see if the problem goes away with using that.
But now we have narrowed the issue down. I think that will be helpful but no idea where to go from here.