Based on my own amateur experiences in making lap dulcimers and
ukuleles, I would suggest 1/8-inch (3 mm) as a maximum thickness.

A wide variety of woods are used on both dulcimers and ukes. Mahogany,
koa, and spruce are very common. For my efforts I have mostly used thin
birch plywood. Thin woods in small sizes are readily found in hobby and
craft stores.

Don't ignore cutouts intended for painting and children's projects if
they are large enough.

Wallace Venable