I have a disappointing answer for anyone following this thread, but feel compelled to provide an update. Vince, to answer your questions: the ArcGIS 10 is at SP5--that connection works. The connection that doesn't work is a version 10.2 client connecting to the 10.2 Oracle geodatabase.
I received, from the DBA, a long list of all the permissions (including public role) associated with the user. I then identified the 18 which existed in dev and test, but not in prod. I asked the DBA to remove those 18 from dev so I could see if their revocation would result in the same error message.
The DBA wrote me back, said I was ready to test, and Eureka! The dev connection works and, mysteriously, so does prod.
Long story short: the DBA fixed the db permissions in prod, so I'll never know which one was the culprit. But I can surmise (but not know for sure) that permissions were the issue.
Interestingly, Vince: neither the user nor the public role are granted execute on dbms_pipe, but the connection works. This seems contrary to the documentation; I'm not sure what to make of it, and since the DBA has hidden the priv tables from me, I can't really get to the bottom of it. Thanks again for your time!