If the people leave for the Milky Way before the end of the reaper war, I don't think they will remember the Normandy as an iconic ship. In ME1, the Normandy was a prototype and participated in the battle of the Citadel, but so did many other warships and later on, more Normandy-class frigates were built so it wasn't one of a kind. The SR2 was first used against the collectors, and not many people even knew about its existence during Shepard's mission, as the whole campaign against the collectors happened with the most of the galactic population unaware. I think the Normandy earned its name as a legendary ship during the reaper war, where it went from planet to planet, forging alliances, helping out people in need etc.