In the first episode of Stargate Atlantis, the only mention of where the 8-symbol address came from, was Daniel Jackson mentioning, "recently, we've been able to determine the...". But how, exactly, did they figure out that the address for Atlantis was 8 symbols, let alone what the 8th symbol was? Was it stored in the Ancient outpost?
This is never revealed
Daniel Jackson's dialogue in Rising, Part 1 is as follows:
We've gotten closer and closer to finding the location of the Lost City but it turns out we've been looking in the wrong place all along. Now, we thought we had a Stargate address -- six symbols representing coordinates in space that determine the location of the planet the Ancients went to after they left Antarctica. Now, recently we determined a seventh symbol.
Unfortunately, the novelization of the episode has the same dialogue and doesn't reveal any more about the circumstances.
When I had originally watched the episode, I had assumed that the incomplete address corresponded to that of a planet related to their search for Atlantis, such as Vis Uban (where SG-1 found a de-ascended Daniel Jackson) or Praclarush (where SG-1 picked up the ZPM to power the Antarctic base). However, the gate addresses don't match, and a search of gate addresses shows that the only two that have even the same first symbol are for unrelated episodes ("A Matter of Time" and "Nemesis").
Given that we don't have any information to go off of, the most likely possibilities in my mind are that the gate addresses came from:
- The Antarctic base where the research team had been working over time to find the address
- The list of gate addresses that Jack O'Neill had placed in the Stargate Command database when he had the Repository of the Ancients downloaded into his head (either time)
- Information that the Asgard had about the last known location of Atlantis
Out of universe, Atlantis was originally planned to be on Earth
Stargate Atlantis was originally going to set after Stargate SG-1 ended after seven seasons, and would have been set on Earth. However when Stargate SG-1 was renewed for an eighth season, it was decided to move the location of Atlantis to another galaxy so that the two shows could run simultaneously without interfering with each other (you can read more about it in the production notes here).
This probably explains the poor setup for this plot point. Since the plan was for it to be on Earth, SG-1 didn't spend all season narrowing down gate addresses or anything. Thus a throwaway line to introduce it being an eight-chevron address in another galaxy was added.
As far as I can tell it was never answered definitively but it was information that was found in between the end of season 7 of SG-1 and the first episode of SGA probably from somewhere in the Antarctic outpost like you said. They knew as of The Fifth Race that an eight chevron address was an intergalactic one, so, they possibly found a reference to it in the outpost's database.