Any tips or advice on convincing her if necessary?
Like any gift you'd like to get from someone it is best if you're explicit.
In the case of family it is also often good to explain that it is as inexpensive a reliably working new item as available, but you wouldn't turn something down that was a bit better if they wanted to spend the extra money.
In the case of a firearm pointing out the utility of it never hurts, but if you live at home you'll have a hard time selling the idea if you include defense. If they thought home defense was an issue they'd already have guns for that purpose and they won't have as easy a time accepting that you having a gun in their home for home defense makes much sense.