Oddly, I have a brayer right in front of me on my office. It is a rubber roller about 4 inches wide and an inch in diameter and has a red handle. I got it to press sticky paper onto a surface to roll out the air bubbles. They are used in printmaking, for rolling ink onto a woodcut block or linoleum cut block. This one is made by Speedball in Statesville, N.C. I found a photo online, see the link below. I bought it at a local art supply store.
A child's rolling pin is probably like like a cooking rolling pin but smaller. We have one at home that came with a "Play Dough" set.
A dowel is just a generic term for a rounded stick of wood, like the bar you put hangers onto in your closet. These come in diameters from about 3/16 inch to 1.5 inches and most hardware stores and hobby stores sell them. A cut-off piece of a wooden broom handle would probably suffice for many uses.