In the next elections, you can choose to stay in your colony, at the risk of having someone else choose to rule it in your stead, or dying, when the colony title reverts to the Head Archon.
The Head Archon is the sovereign of the Freehold and he can only be chosen between the five patrician families, which represent the most impotant dragonlords in the Freehold. So, it basically works as a regular merchant republic but, instead of serving for life, the patricians and other high valyrian characters can get elected to serve as a councillor for for 5 years, after which they choose to receive a colony (a temporary duchy or kingdom title) or a command (a conquest CB that transforms the conquered kingdom in a colony for you to rule). They are sometimes wonky and buggy, but I never really imagined someone could pull this off in the CK2 engine. Now, to the topic at hand: the Freehold mechanics, and how amazing they are (taken from playing the Ghiscary Wars submod). I know that many people that play the mod don't really play vanilla, but I would advise otherwise: it is by comparing the mod with the base game that I came to fully appreciate the amazing work done in it. I still play vanilla regularly, but I always come back to the mod. So, I started playing CK2 back when Legacy of Rome came out and only after reading the Game of Thrones novels I found out about this mod and fell in love with it.