I think that it is very poor navigation if the ship lets the crew sleep much longer than intended and winds up on their home planet instead of the destination they were supposed to reach.
At one percent of the speed of light the ship could have traveled 3 million light years to a nearby galaxy and back in 600,000,000 years. If the mission was to travel for 10,000 years to a star 100 light years from Earth, wake the crew and have them explore, put them back in cold sleep, and return to Earth after another 10,000 years, and the ship instead traveled for 600,000,000 years (30,000 times the planned duration) before returning to Earth, shouldn't the ship have woken the crew and asked their advice or orders about the "minor" change in the mission.
The travelers should ask the navigation computer where they are and be told it is Earth in the distant future. The ship would have calculated and kept track of Earth's ever changing position and landed the ship back on Earth after 600,000,000 years.
Couldn't the navigation computer have been aiming merely to reach any star and planet in our galaxy, and only stumbled upon Sol and Earth by accident and thus not know where it was? That is theoretically possible, but with hundreds of billions of stars in our galaxy, the odds against it would be "astronomical".
It seems to me that they will soon deduce that the universe is 600,000,000 years older. The expansion of the universe will mean that it will be slightly thinner. The galaxies will be spread out more. Deep field images of the farthest visible galaxies will show significantly fewer per square arc second. The nearby galaxies will all be father away, except for the very nearest that are gravity bound to our galaxy and will be measurably closer.
Our galaxy is like a gigantic solar system in that every astronomical body orbits around the center of the galaxy. Some orbit it with periods of a few thousand years, some orbit it with periods a million times that long. Most present open star clusters will have been dispersed by the gravity of passing stars and only a few massive present ones will remain among the newer ones in the future.
But globular star clusters can hold together for many billions of years. They often orbit the galactic center at distances several times that of the The Sun.
The Sun is estimated to take about 225,000,000 to 250,000,000 years to orbit around the galaxy once. In exactly 600,000,000 Earth years the sun should have orbited about 2.4 to 2.66 times.
Globular clusters with 400 million year orbits will have orbited 1.5 times and will be in the opposite points of their orbits. Globular clusters with billion year orbits will have orbited 0.6 times. And so on.
If external galaxies can be identified well enough to show that the spaceship are in our galaxy, the positions of the globular star clusters in their orbits in the external galaxies should tell how far into the future they are.
The angles between where they are, the center of the galaxy, the center of the Andromeda Galaxy, the centers of the Magellanic Clouds, etc, should show them where they are in our galaxy. And they may note that in 600,000,000 years our solar system should have orbited to a similar position in our galaxy.
If they take samples of rocks and minerals from the Earth, Moon, Mars, etc. they should deduce that their chemical composition and isotope ratios are consistent with those planets 600,000,000 years in the future when the radioactive isotopes have decayed for 600,000,000 more years.
If they dig up fossils on the Earth, they will discover that the geologic layers 600,000,000 years old have fossils identical with recent Earth organisms. If they analyze the DNA of the life forms they will discover it has the same base pairs as on Earth. And if they compare the genetic codes of the future Earth live forms to those of present Earth life forms they will find it is consistent with the number of genetic mutations in 600,000,000 years.