According to the wiki X6-88 becomes available as a companion "(…)a short while after the incident is resolved(…)" where I presume the incident is the quest Synth Retention. However I've done the quest some time ago (several in-game weeks) and yet when I speak to X6-88 there is no dialog. I'm currently on the first part of The Battle of Bunker Hill and have no plans to continue on this quest line until I've completed more quests for all the factions.
Fallout – When does X6-88 become available as a companion
fallout-4
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Best Answer
Basically after completing the mission Synth Retention, Father informs you that he has acquired some accommodation for you and that you should go and check it out. X6-88 is waiting outside your accommodation and you can speak to him and ask him to follow you from that point.