You don't play the Humans from the earth... the whole space is randomised. You play Humans from some random planet... most likely.
To be honest that would piss me off. I don't think it's necessary to model our Solar System in the same way every single time (that would be boring as far as starting resources go, but that would also be neat as a toggle : race-specific starting composition, with procedurally generated species choosing one out of a list) but I think it kills immersion when you play Humans from Madachar IV or something. I think it should be possible to name our star and homeworld when creating our race, and I think Humans should get a geographically authentic Earth like they do (I think) in GalcivIII. Of course, for them it'd be Sun and Earth...or "Soleil" and "Terre" if you want to roleplay an Earth where Napoleon won at Waterloo and crushed everyone to establish French hegemony, "Sol" and "Terra" for a Roman Empire in space, etc.
I'd like to see Paradox include a few phenotype-specific lists (so that different species within the phenotype may get different lists) randomly attributed to provinces inside planets, with baseline Humans on Earth having geographically plausible names. As the other species are invented, obviously geographically plausible names are impossible, but I think it'd help players connect with the alien species if they see there is some regularity there. Maybe the player could choose to use a specific list on species creation. Who knows?
I'd also like to see a system where stars you discover have scientific names (astronomical notation) assigned to them when you first stumble into them (or maybe when they are still unexplored- I don't know how fog of war will work in Stellaris given that we can observe the sky with sophisticated equipment now) and you're allowed to choose a name for it after some time (like Betelgeuse, Alpha Centauri, whatever).
I think in sci-fi games more than any other, small bits of flavour like these are extremely important, as there is no historical record to fall back on to produce the feeling that the Universe is believable and real. GalCivIII definitely did not do it correctly (stars named "Gamerz", "Ngenious", "Norway Gladiator", "Rebeccas Jim", "Red Skittles", "Gcookie" or "Godsprogrammer" - Kickstarter backer names) and I felt that without removing all the terrible names the game took a plunge towards very unenjoyable because I was reminded left and right that I was only playing a game.
I also think it's very important for human players to keep a direct link to their planet and history. That's the anchor in history any sci-fi game should have, so you really feel like everyday humans discovering FTL and exploring the galaxy. Their position in the galaxy and the composition of the Solar System might not be perfectly exact (if the option isn't toggled), but that's better than nothing.
A sci-fi game needs a solid setting in order to be engrossing. It needs an anchor in history, which is why I'm excited that all races are going to get a history. History is more than facts however, it's names for geographical features, stars, planets, constellations. If some sort of naming convention can be implemented then Stellaris should be the most engrossing sci-fi strategy game ever created.