Although Leathal Carrot's answer is correct, the women was not asking Arya to take the horse toy, but her daughter, doesn't mean that there wasn't any symbolism and/or possible callbacks with the toy horse, the girl and mother, and/or a thematic connection to the real horse that turns up later.
Knighthood/A Savior
As Ankit Sharma points out in the comments, it could be a symbol of "knighthood", as this scene may call back to Margery Tyrell's scenes where she was passing out wooden horse and knight toys to the poor young children of King's Landing, saying that even if their fathers weren't "anointed" Knights, they fought bravely do defend and protect the city making them out to be just like Knights. She also promises to take care of all of them.
Boy: He wasn’t a knight. He was just a soldier.
Margaery: And what do knights do? Protect the weak and uphold the
good. Your father did that. Be proud of him.
So it's not that Arya is knight in the anointed sense, but she behaved like knight by putting revenge and even herself aside, to try and help others, such as mother and child escape. (This idea is also juxtaposed by Jon killing a Northern soldier, whose about to rape a women).
Foreshadowing Revelations Allegory - Jesus Ressurection/Forgiveness & Death Rider
The toy horse then also foreshadowed the upcoming scene, in which a pale white horse (actually it's a Gray Arabian covered in ashes. I grew up with them!) appears to help Arya escape the scene, as she seems to be the only survivor in the part of the city, proving herself a master of escaping death once more...
"[I]f Christ hath not been raised, our faith is vain; ye are yet in
your sins” (1 Corinthians 15:17).
In Revelation 6:2-8, John sees a vision of the Four Horsemen of the
Apocalypse. The final horseman John the speaker sees is Death, who is
literally “followed” by Hell (or “Hades,” depending on which version
of the Bible you’re reading from), signifying untold destruction that
comes when Death rides in.
From the King James Bible:
“And I looked, and behold a pale horse: and his name that sat on him
was Death, and Hell followed with him.”
What gets a little funny/complicated is that the horse Arya finds in
King’s Landing is a white horse coated in pale gray ash. This has a
double meaning due to the fact that two out of the Four Horsemen rode
white and pale horses that differed in both color and symbolism.
While Death, the black rider, rode a pale horse (often understood to
be ashen gray or sickly green), fellow horseman Pestilence/Conquest
rode a pure white horse, thus earning the name “the white rider.”
As Inverse continued to point out, it's hard to make this all align perfectly, since there are a couple of different translations and because the horse is covered in ash may give a mixed metaphor, but it doesn't mean this iconic imagery is meaningless and it would seem that "Death Rider" would be more likely what the writers are going for, considering Arya's history and what it may metaphysically entail being a Faceless Man...
Other Mythology:
From earliest times, white horses have been mythologised as possessing
exceptional properties, transcending the normal world by having wings
(e.g. Pegasus from Greek mythology), or having horns (the unicorn). As
part of its legendary dimension, the white horse in myth may be
depicted with seven heads (Uchaishravas) or eight feet (Sleipnir),
sometimes in groups or singly. There are also white horses which are
divinatory, who prophesy or warn of danger.
As a rare or distinguished symbol, a white horse typically bears the
hero- or god-figure in ceremonial roles or in triumph over negative
forces. Herodotus reported that white horses were held as sacred
animals in the Achaemenid court of Xerxes the Great (ruled 486–465
BC),6 while in other traditions the reverse happens when it was
sacrificed to the gods.
In more than one tradition, the white horse carries patron saints or
the world savior in the end times (as in Hinduism, Christianity, and
Islam), is associated with the sun or sun chariot (Ossetia) or bursts
into existence in a fantastic way, emerging from the sea or a
lightning bolt.
What Else Do These Scenes Do?
Callback to Shireen Baratheon & R'hllor/Azor Ahai
Shireen's death was one of the hardest in the series, because she was so innocent and lately there has been friendly reminder of innocence lost in the past couple of episodes. But Arguably given the recurring phrase, "Only Death Pays for Life", her death may have been the cost for Melissandre resurrecting Jon Snow and get back into the series debate over if the characters are dying for something greater than themselves or not?
Shireen's sacrifice, Stannis, Melisandre, and Jon's resurrection also then reminds viewers of the Azor prophecy with possible reincarnation of Azor Ahai, with a threw lint to Arya since Arya killed the Night King with Melisandre's, Beric's, and the Hound's help, but yet she and/or the events of The Long Night didn't meet the full criteria...
Now with Dany breaking down, there is more of a set-up for the prophesy to be fulfilled pointing to Jon being in a position where he may have to kill his beloved Queen, but where does that leave Arya?
It's unclear, but there may be some kind of foreshadowing here that Arya contributes to the stabbing or murder of Queen Daenaryes Targaryen. After all in the TV version, Dany doesn't have violet eyes like her book counterpart, but "green" like actress Emelia Clarks. And Melisnadre's "eye prophecy" does include Arya being involved in the death of someone with "green eyes"...
Callback to Cersei Lannister: Motherhood & The Faith of the Seven/True Identity
Uniquely the women with her child that Arya tries to help is women who has short cropped hair. In Season 6 Cersei gets into it with High Sparrow, leader of the Faith Militant (Faith of the Seven), which results not only in Cersei's cutting of hair and walk of atonement, but Cersei's retaliation with the using Wildfire to blow up the Sept, killing Margery (and Loras) along with many innocence, which lead to her last born child's death (fulfilling Maggie the Frog's TV version prophecy).
There is just a lot of imagery and musical score that calls back to season 6's King Landing's scenes through out this episode, including exploding wildfire, but what it all means may go back to arguments given by the Catholic metaphysical poet John Donne ('For Whom the Bell Tolls') and for Cersie more particularly, it may go back to what the High Sparrow said about the truth about whom we really are despite however we look, as both a pregnant Cersei dies in the arms of her Knight Ser Jaime and the poor woman and her child die following their Knight, Arya.
