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On one day in 1989, 43 infants were inexplicably born to random, unconnected women who showed no signs of pregnancy the day before.

Seven were adopted by billionaire industrialist Sir Reginald Hargreeves

Lucy? Was not.
Like many children her age, she certainly followed the exploits of the Umbrella Academy kids, and thought about what it would be like to be one of them. She didn't know until she was sixteen that she could have been, that was when her mother told her that 'that man' had tried to buy her as an infant.

Of course, by then she'd already been manifesting abilities, ones that her mother did her best to keep track of and teach Lucy how to use. Which was difficult, given as how Lucy was lucky, in that she could see possibilities and quickly calculate probabilities. She tested her limits, like any child, and grew to use those skills to her advantage. Because it wasn't simply probabilities, but things like lines of sight and angles of movement. She could cross a crowded room virtually without being seen simply by moving where and when the fewest number of people were looking.

Her mother also told her that the man had said she was the seventh one to say no, which Lucy has no proof of, but she likes to think it's true, because something like that would mean she was doubly lucky.
[Her canon belongs to other people, her face belongs to Anya Rozova, the rest belongs to me.]