fresh_42 said:
Newton's cradle would be less accurate than a single pendulum because of the losses during those collisions and the spread of time periods between the individual pendulums.
A free swinging pendulum without an energy source can be very accurate but, as with real clocks, the pendulum needs to get energy to stop it running down. Foucault's Pendulum would go for, say 24 hours if you could have access to a stair well or very high ceiling (10m+) and a good knife edge mounting to hang it from. Six months running time is a tall order unless you are a pretty good machinst and can make yourself what would be effectively of fully working pendulum clock. Counting the swings involves taking energy from the system unless you use an elecrical detection system.
Let's face it, this is a very tall order. They had enough trouble making accurate clocks when they had access to highly skilled craftsmen and decades of experience.
A good non-electronic time reference, not reliant upon very good mechanisms is Astronomical observations (say the orbits of the moons of Jupiter) that was a serious condender in the quest for Longitude measurements by navigators. But you can't buy a good enough telescope from a hardware store. A solar clock (armillary sun dial) can give accuracy of less than a minute if there is a clear view of the Sun and the shadow has a long throw. The shadow moves at half the rate of a clock hour hand and can be calibrated with a regular clock. It's time varies a bit over the year because of the Earth's Orbit but that's easy to calibrate out that variation. You just need to set it up correctly. You need to live somewhere sunny, for reliability. You could, perhaps, make yourself a Backstaff, which was the forerunner of the Sextant. With that, you could make astronomical observations to get the time from stars, the Moon or the Sun.
Why not go back and ask 'them' to set realistic targets for this project or call their bluff and ask how 'they' would do it.