Several years back, I saw a post on Made by Joel about a DIY Design Felt Board for Babies. I loved the idea, and immediately knew I wanted to make sets for my library. I had a volunteer cover cardboard with felt, (we used black and dark blue because it was on hand), and cut out basic shapes in white. The high contrast makes it easier for little ones to differentiate between the background and the shape. I deployed them in a storytime and sat back to watch the interaction.
Some babies were content to just tap and slap their hands on the boards. Some grabbed onto some the pieces and curled them up tight in their fists. Some swatted the pieces off and weren't sure what to do next. Others carefully picked off the pieces and put them back on. Lots of pieces went straight into mouths. :) Caregivers let the babies explore, and talked about the different shapes and the texture of the felt. I heard many say how neat this was and thought they could make one at home.
Some of you do crafts with your babies and tots, and this could be a fun idea. Felt does come on bolts in the fabric section of a craft or fabric store, which would be great for a project like this so you're not limited to the size of a felt sheet for your background and can get nice big shapes that are appropriate for developing eyes for the pieces.
In terms of storage, my idea was to tape a Ziploc storage bag on the back of each board with packing tape to store the pieces in.