@Zener - schematic is here
. Not much there to go wrong. The DS1337 datasheet
claims it will keep time down to 1.3V. No external caps or resistors required on the oscillator, either.
If I'm reading this thread right, the watch functions normally apart from the time slowth, correct? That would seem to rule out some sort of firmware issue on the 168.
The only thing left would be some sort of failure on the I2C communication. However, the firmware uses the standard arduino RTCLib and, if I'm reading that code right, you would just get garbage or zeros if the I2C read failed - you wouldn't just keep getting the results of the last successful read. So kind of rules out I2C as a source of 'frozen' time.
I can't see anything in the firmware that could account for 'slow' time.
The 168 does go into power-down when the display isn't running, so there exists the possibility of it waking up into a screwy state. But I can't see any way for that to affect timekeeping on the RTC, so long as the voltage doesn't totally brown-out.