In the different Star Trek series some Trill with a symbiont appear. The symbiont has a name the Trill takes like a family name. Well known examples are Jadzia Dax and Ezri Dax, both having the Dax Symbiont.
However, I don't think it is ever revealed in the series how a symbiont gets his name. Do they choose it themselves when they're in the pools, is it given by the other symbionts in the pools, or is it even the first host that can choose its name?
If available, a canon answer would be preferred. If no canon answer is available, a EU answer is also accepted but please clarify that it is a EU answer.