That city kinda put their foot in their mouth. Once they realized their error they should have STFU about it, now she may have reason to move, though she may move anyway for other reasons. Though they can't tax her lotto winnings, they could(most likely) get taxes from other sources of income, such as investment income or property taxes, etc. That would make them as much, but it would get them more than if she chooses to leave.
As for should she give the money to the city anyway? That's her choice, maybe she would have or maybe she would have, over time, given much more. By turning this into a public issue she may choose not to give 1 red cent.
I kinda agree with others, that the amount in question is small and she would not be hurt by paying that at all, but it's her money and her choice whether to donate it or not. Personally, I'd probably give the money to particular programs or projects. Such as: after school sports/activities, community centres, skateboard park, United Way/Salvation Army/other social org, etc.