Disclaimer: I don't know pre-Koiné Greek that well.

The Achaemenids were named after their first ruler Achaemenes, a Greek approximation of old Persian /haxāmaniš/, which we find in the Behistun inscription. The name was translated by B. Schlerath as "characterized by a follower's spirit". I'm not sure what that's supposed to mean.

Now the etymology of Achaeans (Gr. αχαιοι; "akhaioi") is contested, it has been suggested that it was derived from a Hittite form, but Beekes contests this and suggests it's of pre-Greek origin.

Sources:

• Took Old Persian in college
• Beekes, R. An Etymological Dictionary of Greek (Brill: Leiden, 2010)
• Lubotsky, A. & M. De Vaan: An Overview of Old Persian