How accurate the an attiny timer function
The on-chip systemclock is not very accurate. You can calibrate it to some degree but it also depends on the temperature and load of the MCU. You will need a high quality quartz oscillator as external clock source to reduce it. But it will still have a drift. Switching from 8-bit to 16-bit won't make a difference.

In your case an RTC sounds like the way to go.