Update: after an iOS update (to 6.1.2), which had no effect on this issue, I finally bit the bullet and did a complete reset of the phone. Wiped it out completely, restored to zero...
Nuttin; No effect on the issue, which shows it's got to be hardware, not software.
(side note: restore went 100% smoothly, thankfully!)
As I said previously, I meticulously cleaned the docking port, as well as blew compressed into the headphone jack, fwiw, so I'm down to opening it, which I won't attempt (been there, done that, failed catastrophically, with a previous phone).
All moot, at this point, because as a happy coincidence my company is changing providers, which means new phones for everyone (iPhone 5, here I come!).
Just thought I'd make this last post, FYI.