Its almost certain its a home hub 2 issue, probably the plug top power unit failing.
You really need to try a different router. If you do not have one, then I am sure a replacement can be arranged here, as all of the primary diagnostics have been completed.
If you would like to contact one of the UK based BT Care Team who moderate this forum, they should be able to arrange for a replacement, once they have read this thread. It may save a lot of time with the helpdesk, who will only go through the whole process again
They can be contacted using this link BT Care Team
They normally respond by phone or e-mail, within three working days, however you should get an immediate confirmation, with a tracking number.
Best to log off and have a cup of tea - it is usually working when you log on again! All seems a bit slow and "sticky" though at the moment....it would be nice to know there is a technical process going on which is causing it so we know not to mess around with our computers ourselves.
My last post on this subject was 11/11/12.
Shortly after that date a new HomeHub3 and powere unit was delivered and installed by me.
Since that date my connection has not dropped as far as I can tell.
I assume it was the power unit as there were other posts indicating this as a cause for drop outs around that time.