Symbolism, Imagery, Allegory
We first see the river when Daisy and her cousins spend a carefree day there, splashing and frolicking in the sun. Check it out:
The feeling of the cold water and the hot sun and having the river just flow over your skin like a dolphin wasn't something I had enough words to describe but was the kind of feeling you never forget. (1.13.13)
Even though you know everything's going to go downhill from here the second Daisy says "if there ever was a more perfect day in the history of time it isn't one I've heard about" (1.13.17), this day at the river shores up for us as readers the fact that, before everything terrible happens, Daisy and her cousins enjoy family, love, and home. The river, then, marks just how far their situations fall.
Throughout Daisy and Piper's adventures, the river follows them. Once they arrive at their temporary home in Reston Bridge, they discover that their "very own swimming and fishing river […] was a branch of the same one that the bridge in Reston Bridge went over" (1.16.9). Daisy uses this to perk up Piper and convince her that they'll come up with a plan to make it home—the river reminds both of them of what they stand to gain, to win back.
Later, when they're lost in the woods trying to find their way back home, Piper suddenly hears this symbol of hope and family and finding their way home. Daisy has no idea what she's talking about until "her face suddenly burst into a thousand-watt smile and she said It's the river! I'm sure it's the river!" (1.24.9). So of course they "did a little dance and whooped and laughed and hugged each other […] feeling light-footed instead of just light-headed for the first time in days" (1.24.11). They may not be home yet, but the river assures them they're headed in the right direction, metaphorically speaking.