My guess would be that it means the software itself is reset. Therefore everything would need to be off (no ground power, no APU, no main battery, no onboard battery to maintain the memory of the controller, etc.).

As someone who works on high-frequency computer hardware (where one in a billion is every day or so) I'm familiar with the necessity to avoid any error no matter how remote. Obviously aviation is on the other extreme (frequency is much lower, effects are much much much worse) but this is exactly the kind of lurking thing you test for using stress tests (like initializing counters to something near their limits). Apparently Boeing's strategy was to just wait for the counter to overflow which, while a little worrying, isn't the end of the world (thinking of and testing every stress is extremely difficult, time consuming, and expensive).

Personally this could be more damning of the FAA's testing regimen. Time and event based counters should be a no brainier for overflow testing, but if they don't ask for the results then they can't say it was done.