Adorable little girl, and yes, accidental breedings do happen. Sure glad she found a loving, caring home who doesn't care that she was an accident. Enjoy your beautiful puppy. Gretel would be a cute name for her.
I think Daisy is a cute name for a little dog. She is adorable no matter what you name her.
The reason a lot of mutts are not free anymore is because of the people that would get a free dog and then use it as bait to train their fighting dogs. If I had a (female dog) that got knocked up before being spayed. I'd spay her and alter all the pups and then just charge people for what the fixing cost. That way, you aren't contributing to the overpopulation.
I would call her, Bit ! A little Bit, a Bit of this, a Bit of that ! Regardless of her name I'm sure she will make a wonderful pet. Enjoy her,and enjoy ya'lls Mothers Day !!! Bless you, for giving her a happy home !!