Our earliest records of court eunuchs come from the Assyrians, who called them "sa resi." There is some debate if the sa resi are actually eunuchs, but from a few context clues we can be reasonably certain that it does mean eunuch. Here's an article you might like, it's from the 70s but it's open access, it talks about an instance when "sa resi" was used to mean gelded for an animal, which is pretty much bulletproof to me. We also have statues of sa resi in which they are depicted with very eunuchiod features.

However, the Assyrians use of eunuchs is pretty highly developed, pretty similar to the later Byzantines and the Chinese, indicating the practice is much older than the earliest records. So the earliest use of eunuchs is very likely earlier than the Assyrians, but that's the earliest we've got.