Since I wasn't sure where to begin with the concept of "tiny red hat"--other than the obvious--I poked around the craft store and picked up red felt, embroidery thread, purple ribbon, and some various odds and ends. I have yarn and fabric in nearly every color, so I figured if I needed anything else, I would already have it. As it happened, I did not. To get some ideas, I worked on a couple sketches:
Of course, I had some concern over the best way to clip it in, but given my grandma's hair type and preferences, I opted to put two loops on the bottom so the hat can be bobby-pinned to her hair.
I even tried it out myself: