My idea of Sara and Ava's kid in her younger years is that she has always been a self proclaimed time prankster who has caused a lot of time abberations due to her antics. In fact before becoming Captain of the Waverider she had been arrested for her time crimes by the Time Tribunal. Literally a court of law her mother Ava developed after her retirement as a Legend to basically curb a rise in time pirates. Having their own daughter become one of these rouges was especially shell shocking to Ava since she is the one who helped set up the law system. Sara, while not impressed with her daughter's interference of the timeline, is more understanding since she herself went through her own stage of rebellion in her youth before then literally becoming an assassin to survive. Plus she recongizes the Legends are pretty much constantly breaking rules in order to restore time.
A simple introduction to this idea...
“Laurel Olivia Lance you are once again standing before the Time Tribunal for breaking the timeline. You have caused various abbreviations resulting with a level seven time quake. What is your appeal against these charges?”
Laurel smirked. It had all been harmless fun and the Legends were already “fixing" her interference of the timeline. So Cleopatra had become ruler of the Roman Empire? All thanks to one little tank from the year 2370 she may have loaned to her. Which then led to her conquering the entire globe but still. No big deal right?
“My good Tribunal I feel that I must respond to you with my own question. Don’t the pyramids of Los Angeles look lovely this time of year?”
So basically as I stated before Laurel after her aunt. Olivia is for Oliver. Her initials are literally LOL. Yes I realized what I did. No I'm not sorry about it.