The Long Journey Home

The stowaway is a wanted criminal, either a Raxact pirate or a K'schar doctor that appears randomly on your ship after you visited a star system where a Logos Arbiter showed up to inform you of said criminal. If you visit your lab you may see them lurk around and strike up a conversation to get a quest. However, this is not a quest you want to fulfil as it won't get you the achievement.

Once the alien is on board, jump around the cluster but don't go dock on starports or the like. I personally found Logos Arbiter wanting to scan me at the location of the cluster's Jump Gate. Once scanned, hand over the stowaway and the achievement is yours.