I didn't create my bio before starting the game as I had no idea how viable RP'ing in Elite was, and no real clue as to the lore (though I had Frontier as a kid, and loved it despite being so bad at it I could never really get into it as a proper game - I loved what it represented, and the brief intermissions between being blown up... ). As I've said many a time on here, the moment I found out Sol was permit locked, the defining question for my character/RP became 'why does she want to get into Sol?'. Everything else came from that, and that defined how she ended up in that Sidewinder.
I didn't want to create a special snowflake of a character, or a 'chosen one', as I don't think Elite suits or can accommodate that. As in real life, Elite's galaxy is ultimately indifferent to us, so I wanted a character who would fit into that universe. The more extreme or dramatic a character, the more the game's decidedly non-extreme and non-dramatic tone chafes against it...
I won't go into all the details, but essentially older members of the family who'd served with the Federation navy inspired her as a child, both for the Federation's history, and for astronomy/the cosmos in general. With Earth being the cradle of both the Federation and galactic civilisation, Earth's star was always something she looked out for (when she was living on the night side of her tidally locked world, that is).
A period as a cadet in the citystate's landlocked militia made her realise such an overtly structured and subservient life wasn't for her, and so idealism gave way to pragmatism, choosing to go through further education and into the bureaucratic side of her parent's company, connected to the local mining networks.
She never lost the romanticism for the Federation and Sol/Earth, but conventional life got in the way. It was only once she was a valued corporate asset (surely the goal of any Federation loyal individual. ; -) ), and working on her homeworld's orbital station, that she decided - with a couple of friends - to train for her pilot's license (in my bio I reference a system-shackled training Cobra III). The Federal Navy Auxiliary suited her less regimented outlook, and so she bought a Sidewinder to take her first trips out from the system, with the goal of trading and mission running her way towards a Sol permit.
I won't go into how her story's developed, but now it's a few years later, she's an experienced explorer, has a fleet of twenty four ships, and has just outfitted a Corvette (named after her grandmother, before its eventual refit into an FNA registered warship/support craft) for exploration to visit A* for the first time.
I always enjoyed thinking up why characters in Elder Scrolls games (well, chiefly Morrowind and then Skyrim) got into the circumstances at the start of the game, be it a prisoner being released in a strange land, or a prisoner about to get their head hacked off by an Imperial executioner. Elite is both a better and far worse role-player, given there is no conventional story structure, and almost no ways to really express RP in terms of choices, factions, species, nationalities, etc. But if you embrace its freeform, adaptive potential for role-playing, then it can be quite satisfying. I, sparingly, write narrative vignettes to punctuate main events and pivot points in her arc, which is something I've not done since Morrowind first came out, funnily enough. Elite certainly requires more effort to RP, but I've actually enjoyed how detailed and nuanced it's become, and the updates and BGS changes can give a story a shot in the arm or, potentially, a new direction.
So yeah, for the more creative of players out there, for 'why have I just been released onto Vvardenfell, by Imperial decree?' (or 'why am I in Skyrim about to be decapitated... ') see 'why is my character sitting in a Sidewinder?'. Just a little motivation can provide context for actions/goals, and that can open up all kinds of possibilities for what you do or do not do in the game.
For that, I'm glad Elite doesn't try to impose a reason for why you got your Sidewinder.