I don't pretend to have any idea what is going on with Jesus here. I DO know what is going on with the woman though - she is a desperate mother, and those are shoes I've walked in. She's willing to be humiliated, willing to be talked down to, willing to challenge: this is her last hope. And it works. Her willingness to take a scrap turns into a banquet for her child - and that was all that mattered.
I've walked this hard road as a mother more than once. Sometimes I feel like I'm begging God for something that is right; why won't He just give it over? I don't know. It's not an easy position to be in.
I do know that many times in the Gospels, Jesus praises and responds to particularly persistent women. First, it's His own Mother at the wedding at Cana. He tells the story of the judge finally worn down by the woman who badgers and badgers him. Jesus praises the widow's mite, and here, He is overcome with compassion for this mother who will NOT be turned away.
Mostly, this story reminds me not to give up. Prayer isn't always what it seems at our end, and we cannot fathom the mind of God. Although Winston Churchill did not have prayer in mind when he remanded the British people to Never give in--never, never, never, never, in nothing great or small, large or petty, never give in except to convictions of honour and good sense. Never yield to force; never yield to the apparently overwhelming might of the enemy, he may well have been speaking of prayer. That's what I learned from this unyielding woman from so long ago as well.