I often tell my players that they're welcome to do this. They don't need to know their character inside out before they start, and they can make up details as they go. Those details then become part of their story and history going forward.

It gives the players more flexibility and creates opportunities for them to actively to connect with the story.

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: Job offer to take down a corporations new research project

Character 1: "We're taking this job. I lost my brother to that companies bullshit"

Character 2: "What? I didn't even know you had a brother"

Character 1: "I don't like to talk about it much. He was a corper, tried to walk the straight and narrow, and they still fucked him over. This shit is personal for me. We're taking the job"

--

That was a scene that played out in one of my Shadowrun games. Until that moment, not even the player knew his character had a brother. The back story wasn't going to win any awards for originality, and ultimately, it was just a bit of ad-lib the player came up with on the spot to give them a personal connection, yet it changed the whole feel of the mission, and shaped that characters interactions with the company in question going forward in that campaign.

To me, that's the power of creating characters on the fly. Players can always choose to make things personal or relevant to their characters