A flaming brain is when you take a single large sheet of newsprint, and fold the four corners in so that they all meet in the center. The edges of the sheet are now loosely taped together in a few places along all four lines radiating from the center where the edges touch, and the contraption is then "opened out" to form a hollow ball of newspaper, as close to spherical as possible. Then, holding the contraption so that the corners taped together are on the bottom and what used to be the center of the sheet on the top, the paper is ignited right at the bottom where the four corners meet. If done properly, the paper will become a primitive and short-lived hot air balloon, the hot gas and updraft from the flames lifting the newspaper into the air, for a few seconds before the rest of the newspaper is consumed by the flames.
So named because of the appearance (grayish) and shape of the newspaper.
Even a light breeze may overwhelm the effects of the flames on the newspaper as well as blow hot ashes to undesired locations, so for safety's sake as well as performance, it should be done when the wind is calm and nothing that is likely to be accidentally set on fire is nearby.
Joe: I'm done with the newspaper.
Mike: Oh, you want to take a page and go down to the river and do a flaming brain?
Joe: Nah it's too windy