Several people have incorrectly identified her as his daughter - barring some extra-canon material, Max only had a son (Sprog)
who died in the first movie.
Is there any coverage in the extra-cinematic material for this?
The little girl is Glory, a girl Max failed to save from Immortan Joe's son, Scrotus. This is covered in the new Mad Max game.
Essentailly Max gets attacked by Scrotus' War Boys and gets involved in toppling 1 of Immortan's sons. Thus they are hunting for him and capture him in the beginning of Fury Road.
During the events of the game, he comes across a wife named Hope, who is hiding her daughter Glory. After nearly killing Scrotus once he makes it his personal mission to find Max and anyone connected with him.
A Blackfinger named Chumbucket gets tortured for being Max's mechanic and informs Scrotus' top dog, Stank Gum of Max helping Hope and Glory, and where they are hiding. Before Max can get back to them, Scrotus hangs Hope, and leaves Glory to bleed to death.
As for those arguing that the game takes place after and Scotus replaced People Eater, this is incorrect. People Eater was captured by Immortan Joe, and told him of the Citadel to stay alive. Joe appointed him to help run the financing of Gas Town, but Scrotus was in charge, until his death.
Proof of this is found in the game when Hope has Max recover the War Rig due to an earlier failed attempt by former wives.
The specific example I give in my answer is that in Mad Max 2: The Road Warrior and Mad Max 3: Beyond Thunderdome, the same actor (Bruce Spence) is cast in very similar roles (Gyrocopter pilots) but as specfically different characters. This is confusing when watching the films back to back, but ultimately it doesn't really matter - this one small detail doesn't actually matter to the story being told.
George Miller as a director simply isn't interested in the continuity of these films so much as he is simply telling a story, which is the reason that the Interceptor is destroyed in both Mad Max 2: The Road Warrior and in Mad Max: Fury Road without any explanation being given - there simply isn't one. As such, it is best to take the events of these films as if they are campfire stories, myths of the wasteland that may or may not have happened how they are told and all attributed to this mystery wanderer named Max, a post-apocalyptic legend.
As such, it seems safe to say that either the little girl weighing on Max's mind in Mad Max: Fury Road is a reinterpretation to the events of Mad Max when he loses his child to Toecutter and his gang, or she is simply another soul that he failed to save at some point since. Either way, in George Miller's eyes the story being told is more important than the story that has been told, and solid information about the exact events in Max's life are unlikely to appear anytime soon.
btw, we'll know for sure in the near future.
The comic book Mad Max: Fury Road – Mad Max issue #2 will be available on August 26.
The second issue is the more intriguing story though. According to the official synopsis, Max has finished building his V8 Interceptor, only to have it stolen by the Buzzards (the Russian tribe with the spiky cars). To get it back, he heads into the “sunken city.”
Here’s where it gets interesting. The synopsis adds, “But the stakes are much higher than the fate of Max’s ride—an innocent girl, Glory, has also been kidnapped by the depraved Buzzards. By the time Max arrives, it may already be too late…”
Could this Glory be the little girl we see in those hallucinations in Fury Road?
The answer was in the game. The mad max video game takes place just a little before fury road.
<< spoilers for the gamers >>
The little girl is glory. Her mother was hope. I believe they are both lightly mentioned in the movie but never revealed names and the like cause it would be a spoiler to the game. Hope and glory are brutally murdered by lord scrotus. The logo is also the one carved into the side of the mountain in the movie. Idk if I agree with the way they did hope and glory. They should have clarified more in the movie because not everyone is going to play the prologue game.
IMDB lists her as "Glory the Child," so while it's interesting to think of her as a young Furiosa, it doesn't really make any sense (and isn't true anyway). Here's the iMDB cast list.
There's actually some rumoring running around that this "MAX" in fury road isn't even the original. Some specuulate he's the feral boy from 2 grown up as evidenced by the fact he has the music box the real Max gave him in the older films.
With the amount of easter eggs hidden in this movie already, I wouldn't put it past Miller one bit.
His flashbacks from his attempted escape from the citadel show a plethora of souls that he hallucinates whom he couldn't save. I think (a) the little girl is one of the most poignant memories of a soul he couldn't protect during his travels or (b) his mind has become so distraught and deranged that she is an amalgam of his wife Jessie and son Sprog into one single young girl as his suppressed memories created a composite of the two
I think that the little girl was furiosa. She mentioned she was captured as a young girl, max had a chance to help her then but he didn't. Max realizes that furiosa was the girl when they ride over the salt flats. He knows he will be tormented forever if he doesn't help her. Max then helps furiosa to save his own sanity.