If you're going for such drastic change to get rid of the occasional minor issue with leap seconds, then a star clock is a bad idea - the stars move relative to us and each other. The constellations we look upon are differently arranged to the ones Julius Caesar & Co looked upon. You're basically swapping one source of error for another.
Similarly - the oddity of choosing a planet-based time system for the synchronisation of clocks moving interstellar distances? How do they accurately measure time when they're no longer on the planet? And, as others have mentioned, the reason why we have leap seconds in the first place is because the length of a day (and of a year) changes.
It's also worth noting that when stonehenge was used to mark the time, webpages came in the form of bardic tales. If your bard was asleep, you get a 500 error... and they were asleep a lot. Stonehenge time was terrible for information delivery :)
Time is relative, you'll have the feeling of angst until you accept the relativity. No pun intended.
mostly this is tektonic re-adjustments on a minor level or asteroid strike on a major level. of course you could say that an asteroid strike would be a larger problem than re-syncing the stone clock
https://s-media-cache-ak0.pinimg.com/originals/f3/f7/7e/f3f7...
It's illustrative about how hard time is that you tried to create a new system from scratch, with the express purpose of being future proof for space travel, and it's already broken because the fixed point you chose is not, in fact, fixed.
Edit: grammar
I propose a new "non-time" time system. It has exactly two real values which range from 0 to tau and an integer, the first real number is radians of earth rotation, and the second is radians of the rotation around the Sun. The integer reflects the number of complete cycles. So lunch time in Greenwich 'pi'.
It has the benefit that its "source" is actually the planet, so we can use a telescope at Greenwich to pick a certain alignment of stars as the "zero", "zero" point and then each time it realigns to that exact point, you can increment the "year" count.
I believe we can build a robust system to support this out of stone. We'll need to create a circle of stones but using a small hole drilled through a stone and a marker on the ground we can always identify 0.0,0.0, 0.0,pi/2, 0.0, pi, and 0.0, 3*pi/2.