Out of universe, JKR confirmed the name in a cast interview (as well as giving a bit of extra back-story about her confrontation with John Dawlish):
SU: You know, Jo, he idolizes a man, an auror guy who got pwned by an
old lady wearing a dead bird on her head, you know, on her hat. Now,
JN: They don't understand this, Jo. I know we've talked about this.
They don't understand the night that this-- They call it a duel. It
wasn't a duel. At least in my mind anyway, I think it needs a little
explaining of how somebody as, you know, as skillful as Dawlish, you
know, could've got taken down like this.
JKR: By the time Augusta Longbottom got to him, he had been-- several people had attacked Dawlish. I mean, I think he was a bit
punch-drunk by that point, you know. He had become a favorite
punch-bag of the Order of the Phoenix by then. So I don't think he was
firing on all cylinders. But I really saw Mrs. Longbottom as a
powerful witch. So, um, sorry. - Pottercast Interview
The actress portraying the character in the film also explicitly named her;
Our friends at Snitch Seeker have confirmed that actress Ninette Finch was cast as Neville Longbottom’s grandmother in Deathly Hallows.
Finch was also involved in Half-Blood Prince as an extra.
”Yes I am playing Augusta Longbottom, although it is not a very big part… I have also played a couple of witches and was one of the people on the Millennium bridge when it was under attack,” Finch said.