The short story is that the issue was caused by the upgrade of the Ruby driver to 1.4.1. The details are filed in https://jira.mongodb.org/browse/RUBY-347.
At first, we went down the wrong path of believing that this was a networking issue. We were hitting the Mongo server directly and seeing slow database queries as well, but these were caused by the live system opening a large number of connections and the database server not being able to serve more. The number of connections was the first real useful piece of information.
After seeing this graph we backtracked code changes and saw that mongo-ruby-driver (via bson_ext) was upgraded to 1.4.0 right before this started. The changelog for the next version, 1.4.1 said it fixed a number of issues with replica set refresh. A specific commit described a connection leak [6a7e991]. Upgrading to 1.4.1 was an improvement, but the performance issues continued at a lesser extent. Finally, downgrading to mongo/bson/bson_ext 1.3.1 got the system healthy again. The number of connections flattened back down and database performance was back to normal.