Try this link: https://www.youtube.com/watch?v=wBRVB-S8QjM
I believe this is exactly what you need for you project. Professional pool table quality wool green baize would be a perfect material for the interior lining. You would just have to buy some supplies to complete the project listed in the video like a hot glue gun and some stiff card stock sheets for making a template to cut the material to an exact measurement. A few other items like a rotary cutter, X-acto knife and perhaps a cutting mat and of course the box itself but you may already own the latter. You could use the same technique to line the underside of the lid as well and any box you obtain containing a hinged lid with exact dimensions using this woman's method. Good luck on your project.
There are plenty of used boxes being sold that can be used as a chess box. Unfortunately, all of them are not lined. I googled "lining a box" and have plenty of hits with varying lining methods. Is there anyone who has experience lining a box and which method or website was used? Thanks.