A grindstone is used for sharpening tools. On a farm it would be used to sharpen plow blades, axes, etc.
Maybe you've heard the phrase "nose to the grindstone." Putting your own nose to the grindstone means to focus and work really hard. The result, hopefully, will be a polished or sharpened product. If you put someone else's nose to the grindstone you force them to do the hard work that you want them to do.
The grindstone is only mentioned at the beginning of "The Scarlet Ibis," but it's a rather important symbol. Here is the first grindstone quote:
A grindstone stands where the bleeding tree stood, just outside the kitchen door. (1.2)
Wow. So what happened to the bleeding tree, the tree where the ibis was found? (A "bleeding tree" is a tree giving off sap, which is sometimes reddish in color.) Unless it was removed and planted elsewhere, then it died, like the ibis, and like Doodle. OK, so that's interesting but not necessarily significant yet. We need another quote:
But sometimes (like right now), as I sit in the cool, green-draped parlor, the grindstone begins to turn, and time with all its changes is ground away – and I remember Doodle. (1.2)
Ah-ha. Now we're getting somewhere. As we discuss in "What's Up With the Title?" Brother sees a parallel between the ibis dying in the bleeding tree, and Doodle dying "beneath a red nightshade bush beside the road" (4.48). Since the tree used to be "just outside the kitchen door" it would have reminded Brother of Doodle every time he saw it. The grindstone doesn't simply take the tree's physical place, but also its function as a reminder of Doodle, because it stands where the tree stood.
This second quote also suggests that Brother thinks of his mind as a grindstone. When he starts thinking, "time with all its changes is ground away." Brother is also suggesting that memory and forgetting are linked. He has to forget "time with all its changes" to live in the past with Doodle.
The grindstone of his mind polishes and sharpens Brother's memory. And Brother's memory seems polished, indeed. Everything is carefully arranged, sharp, and vivid. There are no moments of confusion, moments where Brother doesn't remember exactly what happened, or what the sky looks like. Literally speaking, the grindstone can't grind away "time with all its changes." In other words, Brother's memory is colored by his grief. He is going to remember his time with Doodle differently if Doodle wasn't dead.
At the same time, his mind does grind away time, because it places Brother back in that time. In his mind, he is reliving that experience. The grindstone analogy works nicely because for Brother, the act of remembering is hard work, and it's painful. The more his mind revolves, the sharper the memory becomes. Like the knives and blades the grindstone is used to sharpen, Brother's memory has the ability to cut him, metaphorically. At the same time, remembering is productive. It produces emotions and feelings in Brother, and allows him to deal with and order his guilt and grief.