If your character is a 21st century astronomer and has a good memory, it should be easy to locate locate Earth from a position within the same galaxy.
If he has a good memory for figures, he can remember the coordinates of various astronomical objects. You may object that all the coordinate systems show the angles of various astronomical objects as seen from Earth and the Solar system.
Yes, but by comparing the angles to various astronomical objects from whatever world he is in now to the coordinates he remembers, he can work backwards to the location of Earth.
If he remembers the coordinates of the center of the galaxy as seen from Earth, and the coordinates of the Andromeda Galaxy as seen from Earth, he will know the angle between them as measured from Earth. So it will be easy to calculate the approximate region where the angle between the Andromeda Galaxy and the center of our galaxy will be the same as seen from earth.
And the same goes for the Large Magellanic Cloud, the Small Magellanic Cloud, the Triangulum Galaxy, the Galaxy M87 at the center of the Virgo Cluster of galaxies, etc. Remembering a bunch of those coordinates should narrow down the region where Earth is to a comparatively small region of the galaxy.
Then If the astronomer from the 21st century remembers the coordinates of a few of the globular star clusters that are tens of thousands of light years from Earth, it can be calculated where within the comparatively small region of the galaxy calculated before is a much smaller region where the angles between those Globular clusters and the galactic center would be correct.
And if the astronomer from the 21st century remembers the coordinates of a few very bright stars as seen from Earth that could help narrow it down. Most of the stars that appear very bright as seen from Earth appear bright as seen from Earth because they are actually very bright or because they are very close to Earth, and usually because of a combination of both factors.
So if the future civilization still uses the 21st century names or designations of those bright stars, the 21st century astronomer will know that Earth is in the small region of the galaxy where those stars are located.
And if the old names are forgotten, the 21st century astronomer can still make use of them once the location of Earth has been narrowed down to a comparatively small region of the galaxy.
Here is a list of the 93 stars that appear brightest as seen from Earth.
https://en.wikipedia.org/wiki/List_of_brightest_stars1
Note how greatly their distances from Earth vary - from 4.4 light years to 2,600 light years. The stars that appear very bright despite being thousands of light years away will be among the brightest stars anywhere within a volume of space several thousand light years across, because they will be among the most intrinsically luminous stars in that volume of space.
They include:
Deneb, Alpha Cygni, 2,600 light years from Earth.
Alnilam, Epsilon Orionis, 2,000 light years from Earth'
Aluda, Eta Canis Majoris, 2,000 light years from Earth.
Wezen, Delta Canis Majoris, 1,800 light years from Earth.
Sadr, Gamma Cygni, 1,500 light years from Earth.
Naos, Zeta Puppis, 1,100 light years from Earth.
As seen from Earth, Alnilam, in Orion, is nearly on the opposite side of Earth from the galactic center, while Deneb and Sadr, in Cygnus, are at about right angles to the line between Orion and the galactic center.
Another guide is the Pleiades star cluster, which is only about 400 to 450 light years from Earth; a less bright star cluster, the Hyades, is only about 150 light years from Earth and in almost the same direction as seen from Earth. So following a line from the Pleiades through the Hyades will lead to a position near Earth.
Once the volume is narrowed down enough, they can search for the brightest star in that volume, which should be Gacrux, Gamma Crucis, over 800 times as luminous as the Sun. Gacrux is about 88 light years from the Sun, but in ten thousand years the distance between the two stars could have changed by as much as about 12 light years. So calculate an inner sphere and an outer sphere around Gacrux, the inner sphere 76 light years from Gacrux, and the outer sphere 100 light year from Gacrux. The Sun should be in or near the space between the two spheres.
Then if you can identify the next brightest star in that volume it might be Aldebaran, Alpha Tauri. Aldebaran has about 518 times the luminosity of the Sun and is about 65 light years away, which could be about 53 to 77 light years in 10,000 years. So chart two spheres around Aldebaran and with radii of 53 and 77 light years, and Earth should be in or near the space between the two spheres.
There should be two volumes where the hollow shell around Gacrux intersects the hollow shell around Aldebaran, and Earth should be in or near one of those two volumes.
And a line from the center of the Hyades star cluster through Aldebaran should point to a spot near Earth.
Thee should be only a few stars nearby nearly as bright as Gacrux and Aldebaran. If one of them is identified as Regulus, Alpha Leonis, for example, that will be a good clue. Regulus has about 288 times the luminosity of the Sun and is about 80 light years away, which could be about 68 to 92 light years in 10,000 years.
So chart two spheres around Regulus and with radii of 68 and 92 light years, and Earth should be in or near the space between the two spheres.
There should be two volumes where the hollow shell around Gacrux intersects the hollow shell around Regulus, and Earth should be in or near one of those two volumes. And there should be two volumes where the hollow shell around Aldebaran intersects the hollow shell around Regulus, and Earth should be in or near one of those two volumes.
And there should be only one volume of space where the hollow shells around Gacrux, Aldebaran, and Regulus interest, and Earth should be in or near that volume of space that contains only a few dozen stars.
Of course, the Sun is unusual in being both a single star without companions, and being much brighter than most stars, so only a few single stars in or near that volume of space should be brighter than the Sun. And astronomers now believe that only about 10 percent of star systems will be similar to the solar system in the arrangement of planets.
And of course the masses and orbits of the four giant planets in our solar system should be basically unchanged by anything that natural forces or human super science could have done in a mere 10,000 years.
And possibly the professional astronomer, amateur astronomer, space enthusiast, science fiction writer, 10-year-old science fiction fan, or whoever, from the 21st century will be transported to 10,000 years in the future with a portable computer of some type with a few astronomy programs of various types so he doesn't have to remember all the details, so long as he can find a way to recharge the batteries.
Since the data files will not be compatible with the computer systems of 10,000 years in the future his portable computer could become the most valuable object in human space as various groups seek to find Earth first.
At least that is the way I think Earth could be located. Other posters here, astronomers or science fiction writers, may have other ideas.
You can also check my answer here:
How would an astronaut conclude he's on Earth, but 600 million years in the future?2
to a similar question.
