Any suggestions? What would you do?
I think it would drive me nuts not knowing what it was and not being able to stop it.
I think it would be a good idea to start a new thread with a link to this one. This one is marked 'solved' and won't get the attention it deserves, whereas a new thread should and some others on here may have better ideas.
Before anything and if you haven't already, make sure you have full backup by way of Time Machine or a clone so that you can restore your data in the event of unforeseen problems if you end up doing a fresh install of OS X, or the more extreme Erase & Install.
I'm not optimistic that this would definitely fix it, but I think I'd try re-installing the OS (NOT Erase & Install) using the Recovery HD. This won't erase your data only replace all the system files:
http://support.apple.com/kb/HT4718
If that didn't fix it, I think I would erase the drive and re-install OS X and then selectively restore my data from the clone of the HD I made before erasing. This could be a bit of a palaver, but I'd worry that doing a blanket restore of all the data would simply re-import the problem and I'd be back at square one. Doing it piecemeal and testing would be a bit of a pain though.
All things considered, I think a new thread and wait for some fresh ideas would be the best next step.